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# Metales representativos, metaloides y no metales
## Incidencia, preparación y propiedades del fósforo
La preparación industrial del fósforo consiste en calentar el fosfato de calcio, obtenido de la roca fosfórica, con arena y coque:
El fósforo se destila fuera del horno y se condensa en un sólido o se quema para formar P4O10. La preparación de muchos otros compuestos de fósforo comienza con P4O10. Los ácidos y fosfatos son útiles como fertilizantes y en la industria química. Otros usos son la fabricación de aleaciones especiales como el ferrofósforo y el bronce de fósforo. El fósforo es importante para fabricar pesticidas, cerillas y algunos plásticos. El fósforo es un no metal activo. En los compuestos, el fósforo suele presentarse en los estados de oxidación 3-, 3+ y 5+. El fósforo presenta números de oxidación inusuales para un elemento del grupo 15 en compuestos que contienen enlaces fósforo-fósforo; algunos ejemplos son el tetrahidruro de difósforo, H2P-PH2, y el trisulfuro de tetrafósforo, P4S3, ilustrado en la .
### Compuestos de fósforo y oxígeno
El fósforo forma dos óxidos comunes, el óxido de fósforo(III) (o hexaóxido de tetrafósforo), P4O6, y el óxido de fósforo(V) (o decaóxido de tetrafósforo), P4O10, ambos mostrados en la . El óxido de fósforo(III) es un sólido cristalino blanco con olor similar al ajo. Su vapor es muy venenoso. Se oxida lentamente en el aire y se inflama cuando se calienta a 70 °C, formando P4O10. El óxido de fósforo(III) se disuelve lentamente en agua fría para formar ácido fosforoso, H3PO3.
El óxido de fósforo(V), P4O10, es un polvo blanco que se prepara quemando fósforo en exceso de oxígeno. Su entalpía de formación es muy alta (-2984 kJ), y es bastante estable y un agente oxidante muy pobre. Al dejar caer el P4O10 en el agua se produce un siseo, calor y ácido ortofosfórico:
Debido a su gran afinidad por el agua, el óxido de fósforo(V) es un excelente agente secante para gases y solventes, y para eliminar el agua de muchos compuestos.
### Compuestos de fósforo y halógenos
El fósforo reacciona directamente con los halógenos, formando trihaluros, PX3, y pentahaluros, PX5. Los trihaluros son mucho más estables que los correspondientes trihaluros de nitrógeno; los pentahaluros de nitrógeno no se forman debido a la incapacidad del nitrógeno para formar más de cuatro enlaces.
Los cloruros PCl3 y PCl5, ambos mostrados en la , son los haluros de fósforo más importantes. El tricloruro de fósforo es un líquido incoloro que se prepara pasando cloro sobre fósforo fundido. El pentacloruro de fósforo es un sólido blanquecino que se prepara oxidando el tricloruro con exceso de cloro. El pentacloruro se sublima cuando se calienta y forma un equilibrio con el tricloruro y el cloro cuando se calienta.
Como la mayoría de los haluros no metálicos, ambos cloruros de fósforo reaccionan con un exceso de agua y dan lugar a cloruro de hidrógeno y a un oxiácido: El PCl3 produce ácido fosforoso H3PO3 y el PCl5 produce ácido fosfórico, H3PO4.
Los pentahaluros de fósforo son ácidos de Lewis debido a los orbitales d de valencia vacíos del fósforo. Estos compuestos reaccionan fácilmente con los iones haluro (bases de Lewis) para producir el anión Mientras que el pentafluoruro de fósforo es un compuesto molecular en todos los estados, los estudios de rayos X muestran que el pentacloruro de fósforo sólido es un compuesto iónico, como el pentabromuro de fósforo, [Br-], y pentaioduro de fósforo, [I-].
### Conceptos clave y resumen
El fósforo (grupo 15) suele presentar estados de oxidación de 3- con metales activos y de 3+ y 5+ con no metales más electronegativos. Los halógenos y el oxígeno oxidarán el fósforo. Los óxidos son el óxido de fósforo(V), P4O10, y el óxido de fósforo(III), P4O6. Los dos métodos habituales para preparar ácido ortofosfórico, H3PO4, son la reacción de un fosfato con ácido sulfúrico o la reacción del agua con óxido de fósforo(V). El ácido ortofosfórico es un ácido triprótico que forma tres tipos de sales.
### Ejercicios de química del final del capítulo
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# Metales representativos, metaloides y no metales
## Incidencia, preparación y compuestos del oxígeno
El oxígeno es el elemento más abundante en la corteza terrestre. La superficie terrestre está compuesta por la corteza, la atmósfera y la hidrosfera. Aproximadamente el 50 % de la masa de la corteza terrestre está formada por oxígeno (combinado con otros elementos, principalmente el silicio). El oxígeno se presenta en el aire como moléculas de O2 y, de forma limitada, como moléculas de O3 (ozono). Forma alrededor del 20 % de la masa del aire. Aproximadamente el 89 % del agua en masa está formada por oxígeno combinado. En combinación con el carbono, el hidrógeno y el nitrógeno, el oxígeno es una parte importante de las plantas y los animales.
El oxígeno es un gas incoloro, inodoro e insípido a temperaturas normales. Es ligeramente más denso que el aire. Aunque es poco soluble en el agua (49 mL de gas se disuelven en 1 L a STP), la solubilidad del oxígeno es muy importante para la vida acuática.
La mayor parte del oxígeno aislado comercialmente procede del aire y el resto de la electrólisis del agua. La separación del oxígeno del aire comienza con el enfriamiento y la compresión del aire hasta su licuación. Al calentarse el aire líquido, el oxígeno, con su punto de ebullición más alto (90 K), se separa del nitrógeno, que tiene un punto de ebullición más bajo (77 K). Es posible separar los demás componentes del aire al mismo tiempo basándose en las diferencias de sus puntos de ebullición.
El oxígeno es esencial en los procesos de combustión, como la quema de combustibles. Las plantas y los animales utilizan el oxígeno del aire en la respiración. La administración de aire enriquecido con oxígeno es una práctica médica importante cuando un paciente recibe un suministro inadecuado de oxígeno debido a un shock, una neumonía o alguna otra enfermedad.
La industria química emplea el oxígeno para oxidar muchas sustancias. Una cantidad significativa de oxígeno producido comercialmente es importante en la eliminación del carbono del hierro durante la producción de acero. También se necesitan grandes cantidades de oxígeno puro en la fabricación de metales y en el corte y la soldadura de metales con sopletes de oxihidrógeno y oxiacetileno.
El oxígeno líquido es importante para la industria espacial. Es un agente oxidante en los motores de cohetes. También es la fuente de oxígeno gaseoso para el mantenimiento de la vida en el espacio.
Como sabemos, el oxígeno es muy importante para la vida. La energía necesaria para el mantenimiento de las funciones corporales normales en los seres humanos y en otros organismos procede de la oxidación lenta de compuestos químicos. El oxígeno es el agente oxidante final en estas reacciones. En los seres humanos, el oxígeno pasa de los pulmones a la sangre, donde se combina con la hemoglobina, produciendo oxihemoglobina. En esta forma, la sangre transporta el oxígeno a los tejidos, donde se transfiere a los mismos. Los productos finales son el dióxido de carbono y el agua. La sangre transporta el dióxido de carbono por las venas hasta los pulmones, donde la sangre libera el dióxido de carbono y recoge otro suministro de oxígeno. La digestión y la asimilación de los alimentos regeneran los materiales consumidos por la oxidación en el organismo; la energía liberada es la misma que si el alimento se quemara fuera del cuerpo.
Las plantas verdes reponen continuamente el oxígeno de la atmósfera mediante un proceso llamado fotosíntesis. Los productos de la fotosíntesis pueden variar, pero, en general, el proceso convierte el dióxido de carbono y el agua en glucosa (un azúcar) y oxígeno utilizando la energía de la luz:
Así, el oxígeno que se convirtió en dióxido de carbono y agua por los procesos metabólicos en plantas y animales vuelve a la atmósfera por la fotosíntesis.
Cuando se hace pasar oxígeno seco entre dos placas cargadas eléctricamente, se forma ozono (O3, ilustrado en la ), un alótropo del oxígeno que posee un olor característico. La formación de ozono a partir del oxígeno es una reacción endotérmica, en la que la energía procede de una descarga eléctrica, del calor o de la luz ultravioleta:
El fuerte olor asociado a las chispas de los equipos eléctricos se debe, en parte, al ozono.
El ozono se forma de forma natural en la atmósfera superior por la acción de la luz ultravioleta del sol sobre el oxígeno que hay. La mayor parte del ozono atmosférico se encuentra en la estratosfera, una capa de la atmósfera que se extiende desde unos 10 a 50 kilómetros por encima de la superficie terrestre. Este ozono actúa como barrera contra la luz ultravioleta dañina del sol, absorbiéndola mediante una reacción química de descomposición:
Los átomos de oxígeno reactivo se recombinan con el oxígeno molecular para completar el ciclo del ozono. La presencia de ozono estratosférico disminuye la frecuencia del cáncer de piel y otros efectos perjudiciales de la radiación ultravioleta. Se ha demostrado claramente que los clorofluorocarbonos, CFC (conocidos comercialmente como freones), que estaban presentes como propulsores de aerosoles en latas de aerosol y como refrigerantes, provocaban el agotamiento del ozono en la estratosfera. Esto ocurrió porque la luz ultravioleta también provoca la descomposición de los CFC, produciendo cloro atómico. Los átomos de cloro reaccionan con las moléculas de ozono, lo que provoca una eliminación neta de moléculas de O3 de la estratosfera. Este proceso se explora en detalle en nuestra cobertura de la cinética química. Existe un esfuerzo mundial para reducir la cantidad de CFC utilizados comercialmente, y el agujero de la capa de ozono ya está comenzando a reducir su tamaño a medida que disminuyen las concentraciones atmosféricas de cloro atómico. Mientras que el ozono en la estratosfera ayuda a protegernos, el ozono en la troposfera es un problema. Este ozono es un componente tóxico del smog fotoquímico.
Los usos del ozono dependen de su reactividad con otras sustancias. Puede utilizarse como agente blanqueador de aceites, ceras, tejidos y almidón: Oxida los compuestos coloreados de estas sustancias a compuestos incoloros. Es una alternativa al cloro como desinfectante del agua.
### Reacciones
El oxígeno elemental es un fuerte agente oxidante. Reacciona con la mayoría de los elementos y muchos compuestos.
### Reacción con los elementos
El oxígeno reacciona directamente a temperatura ambiente o a temperaturas elevadas con todos los demás elementos, excepto los gases nobles, los halógenos y algunos metales de transición de segunda y tercera fila de baja reactividad (los que tienen potenciales de reducción más altos que el cobre). El óxido es un ejemplo de la reacción del oxígeno con el hierro. Los metales más activos forman peróxidos o superóxidos. Los metales menos activos y los no metales producen óxidos. Dos ejemplos de estas reacciones son:
Los óxidos de los halógenos, de al menos uno de los gases nobles y de los metales con potenciales de reducción superiores al del cobre no se forman por la acción directa de los elementos con el oxígeno.
### Reacción con compuestos
El oxígeno elemental también reacciona con algunos compuestos. Si es posible oxidar alguno de los elementos de un determinado compuesto, puede producirse una oxidación posterior por parte del oxígeno. Por ejemplo, el sulfuro de hidrógeno, H2S, contiene azufre con un estado de oxidación de 2-. Dado que el azufre no presenta su estado de oxidación máximo, cabría esperar que el H2S reaccionara con el oxígeno. Lo hace, produciendo agua y dióxido de azufre. La reacción es:
También es posible oxidar óxidos como el CO y el P4O6 que contienen un elemento con un estado de oxidación inferior. La facilidad con la que el oxígeno elemental capta los electrones se refleja en la dificultad para eliminar los electrones del oxígeno en la mayoría de los óxidos. De los elementos, solo el flúor, muy reactivo, puede oxidar los óxidos para formar gas oxígeno.
### Óxidos, peróxidos e hidróxidos
Los compuestos de los metales representativos con el oxígeno se dividen en tres categorías: (1) óxidos, que contienen iones de óxido, O2-; (2) peróxidos, que contienen iones de peróxido, con enlaces simples covalentes oxígeno-oxígeno y un número muy limitado de superóxidos, que contienen iones de superóxido, con enlaces covalentes oxígeno-oxígeno que tienen un orden de enlace de Además, hay (3) hidróxidos, que contienen iones de hidróxido, OH-. Todos los metales representativos forman óxidos. Algunos de los metales del grupo 2 también forman peróxidos, MO2, y los metales del grupo 1 también forman peróxidos, M2O2, y superóxidos, MO2.
### Óxidos
Es posible producir los óxidos de la mayoría de los metales representativos calentando los hidróxidos correspondientes (formando el óxido y el agua gaseosa) o los carbonatos (formando el óxido y el CO2 gaseoso). Las ecuaciones para las reacciones de ejemplo son:
Sin embargo, las sales de metales alcalinos suelen ser muy estables y no se descomponen fácilmente al calentarse. Los óxidos de metales alcalinos son el resultado de las reacciones de reducción-oxidación creadas por el calentamiento de nitratos o hidróxidos con los metales. Las ecuaciones para las reacciones de ejemplo son:
A excepción del óxido de mercurio(II), es posible producir los óxidos de los metales de los grupos 2 a 15 quemando el metal correspondiente en el aire. El miembro más pesado de cada grupo, el miembro para el que el efecto de par inerte es más pronunciado, forma un óxido en el que el estado de oxidación del ion metálico es dos menos que el estado de oxidación del grupo (efecto de par inerte). Por lo tanto, se forman Tl2O, PbO y Bi2O3 al quemar talio, plomo y bismuto, respectivamente. Los óxidos de los miembros más ligeros de cada grupo presentan el estado de oxidación del grupo. Por ejemplo, el SnO2 se forma a partir de la combustión del estaño. El óxido de mercurio(II), HgO, se forma lentamente cuando el mercurio se calienta por debajo de 500 °C; se descompone a temperaturas más altas.
La combustión de los miembros de los grupos 1 y 2 en el aire no es una forma adecuada de formar los óxidos de estos elementos. Estos metales son lo suficientemente reactivos como para combinarse con el nitrógeno del aire, por lo que forman mezclas de óxidos y nitruros iónicos. Varios de ellos también forman peróxidos o superóxidos cuando se calientan en el aire.
Todos los óxidos iónicos contienen el ion de óxido, un aceptor de ion de hidrógeno muy potente. A excepción del óxido de aluminio muy insoluble, Al2O3, del estaño(IV), SnO2, y del plomo(IV), PbO2, los óxidos de los metales representativos reaccionan con los ácidos para formar sales. Algunas ecuaciones para estas reacciones son:
Los óxidos de los metales de los grupos 1 y 2 y del óxido de talio(I) reaccionan con el agua y forman hidróxidos. Ejemplos de estas reacciones son:
Los óxidos de los metales alcalinos tienen poca utilidad industrial, a diferencia del óxido de magnesio, el óxido de calcio y el óxido de aluminio. El óxido de magnesio es importante para la fabricación de ladrillos refractarios, crisoles, revestimientos de hornos y aislamiento térmico, aplicaciones que requieren estabilidad química y térmica. El óxido de calcio, a veces llamado cal viva o cal en el mercado industrial, es muy reactivo, y sus principales usos reflejan su reactividad. El óxido de calcio puro emite una luz blanca intensa cuando se calienta a una temperatura elevada (como se ilustra en la ). Los bloques de óxido de calcio calentados por llamas de gas eran las luces del escenario en los teatros antes de que existiera la electricidad. De ahí viene la frase "en el candelero".
El óxido de calcio y el hidróxido de calcio son bases económicas muy utilizadas en la elaboración de productos químicos, aunque la mayoría de los productos útiles que se preparan a partir de ellos no contienen calcio. El óxido de calcio, CaO, se obtiene calentando el carbonato de calcio, CaCO3, que se puede conseguir de forma amplia y económica en forma de piedra caliza o conchas de ostras:
Aunque esta reacción de descomposición es reversible, es posible obtener un rendimiento del 100 % de CaO dejando escapar el CO2. Es posible preparar hidróxido de calcio mediante la conocida reacción ácido-base de un óxido metálico soluble con agua:
Tanto el CaO como el Ca(OH)2 son útiles como bases; aceptan protones y neutralizan los ácidos.
La alúmina (Al2O3) se presenta en la naturaleza como el mineral corindón, una sustancia muy dura que se utiliza como abrasivo para esmerilar y pulir. El corindón es importante para el comercio de joyería como rubí y zafiro. El color del rubí se debe a la presencia de una pequeña cantidad de cromo; otras impurezas producen la gran variedad de colores posibles para los zafiros. En la actualidad, los rubíes y zafiros artificiales se fabrican fundiendo óxido de aluminio (punto de fusión = 2050 °C) con pequeñas cantidades de óxidos para producir los colores deseados y enfriando la masa fundida de forma que se produzcan grandes cristales. Los láseres de rubí utilizan cristales de rubí sintético.
El óxido de zinc, ZnO, era un útil pigmento de pintura blanca; sin embargo, los contaminantes tienden a decolorar el compuesto. El compuesto también es importante en la fabricación de neumáticos para automóviles y otros productos de caucho, y en la preparación de ungüentos medicinales. Por ejemplo, los protectores solares a base de óxido de zinc, como se muestra en la , ayudan a prevenir las quemaduras solares. El óxido de zinc de estos protectores solares está presente en forma de granos muy pequeños conocidos como nanopartículas. El dióxido de plomo es un componente de los acumuladores de plomo cargados. El plomo(IV) tiende a convertirse en el ion más estable de plomo(II) al ganar dos electrones, por lo que el dióxido de plomo es un potente agente oxidante.
### Peróxidos y superóxidos
Los peróxidos y superóxidos son fuertes oxidantes y son importantes en los procesos químicos. El peróxido de hidrógeno, H2O2, preparado a partir de peróxidos metálicos, es un importante blanqueador y desinfectante. Los peróxidos y superóxidos se forman cuando el metal o los óxidos metálicos de los grupos 1 y 2 reaccionan con el oxígeno puro a temperaturas elevadas. El peróxido de sodio y los peróxidos de calcio, estroncio y bario se forman al calentar el metal u óxido metálico correspondiente en oxígeno puro:
Los peróxidos de potasio, rubidio y cesio pueden prepararse calentando el metal o su óxido en una cantidad de oxígeno cuidadosamente controlada:
Con un exceso de oxígeno, se forman los superóxidos KO2, RbO2 y CsO2. Por ejemplo:
La estabilidad de los peróxidos y superóxidos de los metales alcalinos aumenta a medida que aumenta el tamaño del catión.
### Hidróxidos
Los hidróxidos son compuestos que contienen el ion de OH-. Es posible preparar estos compuestos mediante dos tipos generales de reacciones. Los hidróxidos metálicos solubles pueden producirse por la reacción del metal o del óxido metálico con el agua. Los hidróxidos metálicos insolubles se forman cuando una solución de una sal soluble del metal se combina con una solución que contiene iones de hidróxido.
A excepción del berilio y el magnesio, los metales de los grupos 1 y 2 reaccionan con el agua para formar hidróxidos e hidrógeno gaseoso. Ejemplos de estas reacciones son:
Sin embargo, estas reacciones pueden ser violentas y peligrosas, por lo que es preferible producir hidróxidos metálicos solubles mediante la reacción del óxido respectivo con agua:
La mayoría de los óxidos metálicos son anhídridos de base. Esto es obvio para los óxidos solubles porque forman hidróxidos metálicos. La mayoría de los demás óxidos metálicos son insolubles y no forman hidróxidos en el agua; sin embargo, siguen siendo anhídridos de base porque reaccionan con los ácidos.
Es posible preparar los hidróxidos insolubles de berilio, magnesio y otros metales representativos mediante la adición de hidróxido de sodio a una solución de una sal del metal respectivo. Las ecuaciones iónicas netas para las reacciones que involucran una sal de magnesio, una sal de aluminio y una sal de zinc son:
Debe evitarse un exceso de hidróxido al preparar hidróxidos de aluminio, galio, zinc y estaño(II), o los hidróxidos se disolverán con la formación de los correspondientes iones complejos y (vea la ). El aspecto importante de los iones complejos para este capítulo es que se forman por una reacción ácido-base de Lewis, siendo el metal el ácido de Lewis.
La industria utiliza grandes cantidades de hidróxido de sodio como base fuerte y barata. El cloruro de sodio es el material de partida para la producción de NaOH porque el NaCl es un material de partida menos costoso que el óxido. El hidróxido de sodio se encuentra entre los 10 productos químicos de mayor producción en los Estados Unidos, y esta producción se realizaba casi en su totalidad por electrólisis de soluciones de cloruro de sodio. Este proceso es el proceso cloro-álcali, y es el principal método para producir cloro.
El hidróxido de sodio es un compuesto iónico y se funde sin descomponerse. Es muy soluble en agua, desprende mucho calor y forma soluciones muy básicas: 40 gramos de hidróxido de sodio se disuelven en solo 60 gramos de agua a 25 °C. El hidróxido de sodio se emplea en la producción de otros compuestos de sodio y se utiliza para neutralizar soluciones acídicas durante la producción de otros productos químicos, como los petroquímicos y los polímeros.
Muchas de las aplicaciones de los hidróxidos son para la neutralización de ácidos (como el antiácido que se muestra en la ) y para la preparación de óxidos por descomposición térmica. Una suspensión acuosa de hidróxido de magnesio constituye el antiácido leche de magnesia. Debido a su fácil disponibilidad (a partir de la reacción del agua con el óxido de calcio preparado por la descomposición de la piedra caliza, CaCO3), su bajo costo y su actividad, el hidróxido de calcio se utiliza ampliamente en aplicaciones comerciales que necesitan una base barata y fuerte. La reacción de los hidróxidos con los ácidos adecuados también se utiliza para preparar sales.
### Compuestos de oxígeno no metálicos
La mayoría de los no metales reaccionan con el oxígeno para formar óxidos no metálicos. Dependiendo de los estados de oxidación disponibles para el elemento, se pueden formar diversos óxidos. El flúor se combina con el oxígeno para formar fluoruros como OF2, donde el oxígeno tiene un estado de oxidación 2+.
### Compuestos de azufre y oxígeno
Los dos óxidos de azufre más comunes son el dióxido de azufre, SO2, y el trióxido de azufre, SO3. El olor del azufre quemado proviene del dióxido de azufre. El dióxido de azufre, que se muestra en la , aparece en los gases volcánicos y en la atmósfera cerca de las plantas industriales que queman combustible que contiene compuestos de azufre.
La producción comercial de dióxido de azufre procede de la quema de azufre o de la tostación de minerales sulfurosos como ZnS, FeS2 y Cu2S en el aire. (La tostación, que forma el óxido metálico, es el primer paso en la separación de muchos metales de sus minerales). Un método conveniente para preparar el dióxido de azufre en el laboratorio es por la acción de un ácido fuerte sobre sales de sulfito que contienen el ion de o sales de hidrógeno sulfito que contienen El ácido sulfuroso, H2SO3, se forma primero, pero se descompone rápidamente en dióxido de azufre y agua. El dióxido de azufre también se forma cuando muchos agentes reductores reaccionan con ácido sulfúrico caliente y concentrado. El trióxido de azufre se forma lentamente al calentar el dióxido de azufre y el oxígeno juntos, y la reacción es exotérmica:
El dióxido de azufre es un gas a temperatura ambiente, y la molécula de SO2 está doblada. El trióxido de azufre se funde a 17 °C y hierve a 43 °C. En estado de vapor, sus moléculas son unidades individuales de SO3 (mostradas en la ), pero en estado sólido, el SO3 existe en varias formas poliméricas.
Los óxidos de azufre reaccionan como ácidos de Lewis con muchos óxidos e hidróxidos en reacciones ácido-base de Lewis, con la formación de sulfitos o sulfitos de hidrógeno, y sulfatos o sulfatos de hidrógeno, respectivamente.
### Compuestos de oxígeno y halógenos
Los halógenos no reaccionan directamente con el oxígeno, pero es posible preparar compuestos binarios oxígeno-halógeno mediante las reacciones de los halógenos con compuestos que contienen oxígeno. Los compuestos de oxígeno con cloro, bromo y yodo son óxidos porque el oxígeno es el elemento más electronegativo en estos compuestos. Por otro lado, los compuestos de flúor con oxígeno son fluoruros porque el flúor es el elemento más electronegativo.
Como clase, los óxidos son extremadamente reactivos e inestables, y su química tiene poca importancia práctica. El óxido de dicloro, formalmente llamado monóxido de dicloro, y el dióxido de cloro, ambos mostrados en la , son los únicos compuestos comercialmente importantes. Son importantes como agentes blanqueadores (para su uso con la pulpa y la harina) y para el tratamiento del agua.
### Oxiácidos no metálicos y sus sales
Los óxidos no metálicos forman ácidos cuando se les deja reaccionar con agua; son anhídridos de ácido. Los oxianiones resultantes pueden formar sales con varios iones metálicos.
### Oxiácidos y sales de nitrógeno
El pentaóxido de nitrógeno, N2O5, y el NO2 reaccionan con el agua para formar ácido nítrico, HNO3. Los alquimistas, ya en el siglo VIII, conocían el ácido nítrico (mostrado en la ) como aqua fortis (que significa "agua fuerte"). El ácido era útil en la separación del oro de la plata porque disuelve la plata pero no el oro. Después de las tormentas, aparecen en la atmósfera trazas de ácido nítrico y sus sales están ampliamente distribuidas en la naturaleza. Existen enormes depósitos de salitre de Chile, NaNO3, en la región desértica cercana a la frontera de Chile y Perú. El salitre de Bengala, KNO3, se encuentra en la India y en otros países del Extremo Oriente.
En el laboratorio, es posible producir ácido nítrico calentando una sal de nitrato (como el nitrato de sodio o de potasio) con ácido sulfúrico concentrado:
El proceso de Ostwald es el método comercial para producir ácido nítrico. Este proceso implica la oxidación del amoníaco a óxido nítrico, NO, la oxidación del óxido nítrico a dióxido de nitrógeno, NO2 y la posterior oxidación e hidratación del dióxido de nitrógeno para formar ácido nítrico:
O
El ácido nítrico puro es un líquido incoloro. Sin embargo, suele ser de color amarillo o marrón porque se forma NO2 al descomponerse el ácido. El ácido nítrico es estable en solución acuosa; las soluciones que contienen el 68 % del ácido son el ácido nítrico concentrado disponible en el mercado. Es a la vez un fuerte agente oxidante y un ácido fuerte.
La acción del ácido nítrico sobre un metal rara vez produce H2 (por reducción de H+) en más que pequeñas cantidades. En cambio, se produce la reducción del nitrógeno. Los productos formados dependen de la concentración del ácido, la actividad del metal y la temperatura. Normalmente, se forma una mezcla de nitratos, óxidos de nitrógeno y diversos productos de reducción. Los metales menos activos, como el cobre, la plata y el plomo, reducen el ácido nítrico concentrado principalmente a dióxido de nitrógeno. La reacción del ácido nítrico diluido con el cobre produce NO. En cada caso, las sales de nitrato de los metales cristalizan al evaporar las soluciones resultantes.
Los elementos no metálicos, como el azufre, el carbono, el yodo y el fósforo, se oxidan con el ácido nítrico concentrado hasta convertirse en sus óxidos u oxiácidos, con la formación de NO2:
El ácido nítrico oxida muchos compuestos; por ejemplo, el ácido nítrico concentrado oxida fácilmente el ácido clorhídrico a cloro y dióxido de cloro. Una mezcla de una parte de ácido nítrico concentrado y tres partes de ácido clorhídrico concentrado (llamada aqua regia, que significa agua real) reacciona vigorosamente con los metales. Esta mezcla es especialmente útil para disolver el oro, el platino y otros metales más difíciles de oxidar que el hidrógeno. Una ecuación simplificada para representar la acción del aqua regia sobre el oro es:
Los nitratos, sales del ácido nítrico, se forman cuando los metales, óxidos, hidróxidos o carbonatos reaccionan con el ácido nítrico. La mayoría de los nitratos son solubles en agua; de hecho, uno de los usos importantes del ácido nítrico es la preparación de nitratos metálicos solubles.
El ácido nítrico se utiliza ampliamente en el laboratorio y en las industrias químicas como ácido fuerte y agente oxidante fuerte. Es importante en la fabricación de explosivos, tintes, plásticos y medicamentos. Las sales de ácido nítrico (nitratos) son valiosas como fertilizantes. La pólvora es una mezcla de nitrato de potasio, azufre y carbón vegetal.
La reacción del N2O3 con el agua da una solución azul pálido de ácido nitroso, HNO2. Sin embargo, el HNO2 (mostrado en la ) es más fácil de preparar mediante la adición de un ácido a una solución de nitrito; el ácido nitroso es un ácido débil, por lo que el ion de nitrito es básico en solución acuosa:
El ácido nitroso es muy inestable y solo existe en solución. Se desproporciona lentamente a temperatura ambiente (rápidamente cuando se calienta) en ácido nítrico y óxido nítrico. El ácido nitroso es un agente oxidante activo con fuertes agentes reductores, y los agentes oxidantes fuertes lo oxidan a ácido nítrico.
El nitrito de sodio, NaNO2, es un aditivo para carnes como los perritos calientes y los embutidos. El ion de nitrito tiene dos funciones. Limita el crecimiento de las bacterias que pueden causar intoxicaciones alimentarias y prolonga la retención del color rojo de la carne. La adición de nitrito de sodio a los productos cárnicos es controvertida porque el ácido nitroso reacciona con ciertos compuestos orgánicos para formar una clase de compuestos conocidos como nitrosaminas. Las nitrosaminas producen cáncer en los animales de laboratorio. Esto ha llevado a la FDA a limitar la cantidad de NaNO2 en los alimentos.
Los nitritos son mucho más estables que el ácido, pero los nitritos, al igual que los nitratos, pueden explotar. Los nitritos, al igual que los nitratos, también son solubles en agua (el AgNO2 es solo ligeramente soluble).
### Oxiácidos y sales de fósforo
El ácido ortofosfórico puro, H3PO4 (mostrado en la ), forma cristales incoloros y delicuescentes que se funden a 42 °C. El nombre común de este compuesto es ácido fosfórico, y está disponible comercialmente como una solución viscosa al 82 % conocida como ácido fosfórico almibarado. Uno de los usos del ácido fosfórico es como aditivo de muchos refrescos.
Un método comercial para preparar ácido ortofosfórico consiste en tratar la roca de fosfato de calcio con ácido sulfúrico concentrado:
La dilución de los productos con agua, seguida de la filtración para eliminar el sulfato de calcio, da lugar a una solución ácida diluida contaminada con dihidrogenofosfato de calcio, Ca(H2PO4)2, y otros compuestos asociados a la roca fosfórica de calcio. Es posible preparar ácido ortofosfórico puro disolviendo P4O10 en agua.
La acción del agua sobre el P4O6, el PCl3, el PBr3 o el PI3 forma ácido fosforoso, H3PO3 (mostrado en la ). El mejor método para preparar ácido fosforoso puro es la hidrólisis del tricloruro de fósforo:
Al calentar la solución resultante se expulsa el cloruro de hidrógeno y se produce la evaporación del agua. Cuando se evapora suficiente agua, aparecen cristales blancos de ácido fosforoso al enfriarse. Los cristales son delicuescentes, muy solubles en agua, y tienen un olor como el del ajo. El sólido funde a 70,1 °C y se descompone a unos 200 °C por desproporción en fosfina y ácido ortofosfórico:
El ácido fosforoso forma solo dos series de sales, que contienen el ion de dihidrógeno fosfito, o el ion de fosfato de hidrógeno, respectivamente. No es posible sustituir el tercer átomo de hidrógeno porque no es muy acídico, ya que no es fácil ionizar el enlace P-H.
### Oxiácidos y sales de azufre
La preparación del ácido sulfúrico, H2SO4 (mostrado en la ), comienza con la oxidación del azufre a trióxido de azufre y luego la conversión del trióxido a ácido sulfúrico. El ácido sulfúrico puro es un líquido incoloro y aceitoso que se congela a 10,5 °C. Emite vapores cuando se calienta porque el ácido se descompone en agua y trióxido de azufre. El proceso de calentamiento hace que se pierda más trióxido de azufre que agua, hasta alcanzar una concentración del 98,33 % de ácido. El ácido de esta concentración hierve a 338 °C sin más cambios en la concentración (una solución de ebullición constante) y es el H2SO4 comercialmente concentrado. La cantidad de ácido sulfúrico utilizada en la industria supera la de cualquier otro compuesto manufacturado.
La fuerte afinidad del ácido sulfúrico concentrado por el agua lo convierte en un buen agente deshidratante. Es posible secar gases y líquidos inmiscibles que no reaccionan con el ácido haciéndolos pasar por este.
El ácido sulfúrico es un ácido diprótico fuerte que se ioniza en dos etapas. En una solución acuosa, la primera etapa es esencialmente completa. La ionización secundaria no es tan completa, y es un ácido moderadamente fuerte (alrededor del 25 % de ionización en solución de una sal de : Ka = 1,2 10−2).
Al ser un ácido diprótico, el ácido sulfúrico forma tanto sulfatos, como el Na2SO4, como sulfatos de hidrógeno, como el NaHSO4. La mayoría de los sulfatos son solubles en agua; sin embargo, los sulfatos de bario, estroncio, calcio y plomo son solo ligeramente solubles en agua.
Entre los sulfatos importantes están el Na2SO4⋅10H2O y las sales de Epsom, MgSO4⋅7H2O. Porque el es un ácido, los sulfatos de hidrógeno, como el NaHSO4, presentan un comportamiento acídico, y este compuesto es el principal ingrediente de algunos limpiadores domésticos.
El ácido sulfúrico caliente y concentrado es un agente oxidante. Dependiendo de su concentración, la temperatura y la fuerza del agente reductor, el ácido sulfúrico oxida muchos compuestos y, en el proceso, sufre una reducción a SO2, S, H2S, o S2-.
El dióxido de azufre se disuelve en agua para formar una solución de ácido sulfuroso, como es de esperar para el óxido de un no metal. El ácido sulfuroso es inestable y no es posible aislar el H2SO3 anhidro. El calentamiento de una solución de ácido sulfuroso expulsa el dióxido de azufre. Como otros ácidos dipróticos, el ácido sulfuroso se ioniza en dos pasos: Se forman el ion de hidrógeno sulfito, y el ion de sulfito, . El ácido sulfuroso es un ácido moderadamente fuerte. La ionización es de aproximadamente el 25 % en la primera etapa, pero es mucho menor en la segunda(Ka1 = 1,2 10-2 y Ka2 = 6,2 10−8).
Para preparar sales sólidas de sulfito e hidrógeno sulfito, es necesario añadir una cantidad estequiométrica de una base a una solución de ácido sulfuroso y luego evaporar el agua. Estas sales también se forman a partir de la reacción del SO2 con óxidos e hidróxidos. El calentamiento del hidrogenosulfito de sodio sólido forma sulfito de sodio, dióxido de azufre y agua:
Los agentes oxidantes fuertes pueden oxidar el ácido sulfuroso. El oxígeno del aire lo oxida lentamente hasta convertirlo en ácido sulfúrico, más estable:
Las soluciones de sulfitos también son muy susceptibles a la oxidación del aire para producir sulfatos. Por lo tanto, las soluciones de sulfitos siempre contienen sulfatos después de la exposición al aire.
### Oxiácidos halógenos y sus sales
Los compuestos HXO, HXO2, HXO3 y HXO4, donde X representa Cl, Br o I, son los ácidos hipohalos, halos, hálicos y perhálicos, respectivamente. Las fuerzas de estos ácidos aumentan desde los ácidos hipohalos, que son ácidos muy débiles, hasta los ácidos perhalos, que son muy fuertes. La enumera los ácidos conocidos y, cuando se conocen, sus valores de pKa se indican entre paréntesis.
El único oxiácido de flúor que se conoce es el ácido hipofluoroso, HOF, muy inestable, que se prepara por la reacción del flúor gaseoso con el hielo:
El compuesto es muy inestable y se descompone por encima de -40 °C. Este compuesto no se ioniza en el agua y no se conocen sales. No es seguro que el nombre de ácido hipofluorado sea apropiado para el HOF; un nombre más apropiado podría ser hipofluorita de hidrógeno.
Las reacciones del cloro y el bromo con el agua son análogas a la del flúor con el hielo, pero estas reacciones no llegan a completarse, y resultan mezclas del halógeno y de los respectivos ácidos hipohalos e hidrohálicos. Aparte del HOF, los ácidos hipohalos solo existen en solución. Los ácidos hipohalos son todos ácidos muy débiles; sin embargo, el HOCl es un ácido más fuerte que el HOBr, que, a su vez, es más fuerte que el HOI.
La adición de bases a las soluciones de los ácidos hipohalos produce soluciones de sales que contienen los iones básicos de hipohalitos, OX-. Es posible aislar estas sales como sólidos. Todos los hipohalitos son inestables con respecto a la desproporción en solución, pero la reacción es lenta para el hipoclorito. El hipobromito y el hipoiodito se desproporcionan rápidamente, incluso en el frío:
El hipoclorito de sodio es un blanqueador (Clorox) y germicida barato. La preparación comercial consiste en la electrólisis de soluciones acuosas frías de cloruro de sodio en condiciones en las que el cloro y el ion de hidróxido resultantes pueden reaccionar. La reacción neta es:
El único ácido halo definitivamente conocido es el ácido cloroso, HClO2, obtenido por la reacción del clorito de bario con ácido sulfúrico diluido:
La filtración del sulfato de bario insoluble deja una solución de HClO2. El ácido cloroso no es estable; se descompone lentamente en la solución para producir dióxido de cloro, ácido clorhídrico y agua. El ácido clorhídrico reacciona con las bases para dar sales que contienen el ion de clorito (mostradas en la ). El clorito sódico tiene una amplia aplicación en el blanqueo del papel porque es un fuerte agente oxidante y no daña el papel.
El ácido clórico, HClO3, y el ácido brómico, HBrO3, solo son estables en solución. La reacción del yodo con el ácido nítrico concentrado produce un ácido yódico blanco estable, HIO3:
Es posible obtener los ácidos hálicos más ligeros a partir de sus sales de bario por reacción con ácido sulfúrico diluido. La reacción es análoga a la utilizada para preparar el ácido cloroso. Todos los ácidos hálicos son ácidos fuertes y agentes oxidantes muy activos. Los ácidos reaccionan con las bases para formar sales que contienen iones de clorato (mostradas en la ). Otro método de preparación es la oxidación electroquímica de una solución caliente de un haluro metálico para formar los cloratos metálicos adecuados. El clorato de sodio es un herbicida; el clorato de potasio se utiliza como agente oxidante.
El ácido perclórico, HClO4, se forma al tratar un perclorato, como el perclorato de potasio, con ácido sulfúrico a presión reducida. El HClO4 puede destilarse de la mezcla:
Las soluciones acuosas diluidas de ácido perclórico son bastante estables térmicamente, pero las concentraciones superiores al 60 % son inestables y peligrosas. El ácido perclórico y sus sales son potentes agentes oxidantes, ya que el cloro, muy electronegativo, es más estable en un estado de oxidación inferior al 7+. Se han producido graves explosiones al calentar soluciones concentradas con sustancias fácilmente oxidables. Sin embargo, sus reacciones como agente oxidante son lentas cuando el ácido perclórico está frío y diluido. Este ácido es uno de los más fuertes de todos los ácidos. La mayoría de las sales que contienen el ion de perclorato (mostradas en la ) son solubles. Es posible prepararlas a partir de reacciones de bases con ácido perclórico y, comercialmente, por la electrólisis de soluciones calientes de sus cloruros.
Las sales de perbromato son difíciles de preparar, y las mejores síntesis actuales implican la oxidación de bromatos en solución básica con flúor gaseoso, seguida de acidificación. Los usos comerciales de este ácido o de sus sales son escasos o nulos.
Hay varios ácidos diferentes que contienen yodo en el estado de oxidación 7+; entre ellos se encuentran el ácido metaperiódico, HIO4, y el ácido paraperiódico, H5IO6. Estos ácidos son fuertes agentes oxidantes y reaccionan con las bases para formar las sales correspondientes.
### Conceptos clave y resumen
El oxígeno es uno de los elementos más reactivos. Esta reactividad, unida a su abundancia, hace que la química del oxígeno sea muy rica y se comprenda bien.
Los compuestos de los metales representativos con el oxígeno existen en tres categorías (1) óxidos, (2) peróxidos y superóxidos, e (3) hidróxidos. El calentamiento de los hidróxidos, nitratos o carbonatos correspondientes es el método más común para producir óxidos. El calentamiento del metal o del óxido metálico en oxígeno puede dar lugar a la formación de peróxidos y superóxidos. Los óxidos solubles se disuelven en agua para formar soluciones de hidróxidos. La mayoría de los óxidos metálicos son anhídridos de base y reaccionan con los ácidos. Los hidróxidos de los metales representativos reaccionan con los ácidos en reacciones ácido-base para formar sales y agua. Los hidróxidos tienen muchos usos comerciales.
Todos los no metales, excepto el flúor, forman múltiples óxidos. Casi todos los óxidos no metálicos son anhídridos de ácido. La acidez de los oxiácidos requiere que los átomos de hidrógeno se unan a los átomos de oxígeno de la molécula y no a otro átomo no metálico. En general, la fuerza del oxiácido aumenta con el número de átomos de oxígeno unidos al átomo no metálico y no a un hidrógeno.
### Ejercicios de química del final del capítulo
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# Metales representativos, metaloides y no metales
## Incidencia, preparación y propiedades del azufre
El azufre existe en la naturaleza como depósitos elementales, así como sulfuros de hierro, zinc, plomo y cobre, y sulfatos de sodio, calcio, bario y magnesio. El sulfuro de hidrógeno suele ser un componente del gas natural y está presente en muchos gases volcánicos, como los que se muestran en la . El azufre es un componente de muchas proteínas y es esencial para la vida.
El proceso de Frasch, ilustrado en la , es importante en la extracción de azufre libre de enormes depósitos subterráneos en Texas y Luisiana. El agua sobrecalentada (170 °C y 10 atm de presión) se hace descender por el más externo de los tres tubos concéntricos hasta el depósito subterráneo. El agua caliente derrite el azufre. El tubo más interno conduce el aire comprimido hacia el azufre líquido. El aire obliga al azufre líquido, mezclado con el aire, a subir por el tubo de salida. La transferencia de la mezcla a grandes cubas de decantación permite que el azufre sólido se separe al enfriarse. Este azufre tiene una pureza de entre el 99,5 % y el 99,9 % y no necesita ser purificado para la mayoría de los usos.
Las mayores cantidades de azufre también provienen del sulfuro de hidrógeno recuperado durante la purificación del gas natural.
El azufre existe en varias formas alotrópicas. La forma estable a temperatura ambiente contiene anillos de ocho miembros, por lo que la fórmula verdadera es S8. Sin embargo, los químicos suelen utilizar S para simplificar los coeficientes en las ecuaciones químicas; en este libro seguiremos esta práctica.
Al igual que el oxígeno, que también pertenece al grupo 16, el azufre presenta un comportamiento claramente no metálico. Oxida los metales, dando una variedad de sulfuros binarios en los que el azufre presenta un estado de oxidación negativo (2-). El azufre elemental oxida los no metales menos electronegativos, y los no metales más electronegativos, como el oxígeno y los halógenos, lo oxidan. Otros agentes oxidantes fuertes también oxidan el azufre. Por ejemplo, el ácido nítrico concentrado oxida el azufre al ion de sulfato, con la formación simultánea de óxido de nitrógeno(IV):
La química del azufre con un estado de oxidación de 2- es similar a la del oxígeno. Sin embargo, a diferencia del oxígeno, el azufre forma muchos compuestos en los que presenta estados de oxidación positivos.
### Conceptos clave y resumen
El azufre (grupo 16) reacciona con casi todos los metales y forma fácilmente el ion de sulfuro, S2-, en el que tiene como estado de oxidación 2-. El azufre reacciona con la mayoría de los no metales.
### Ejercicios de química del final del capítulo
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# Metales representativos, metaloides y no metales
## Incidencia, preparación y propiedades de los halógenos
Los elementos del grupo 17 son los halógenos. Son los elementos flúor, cloro, bromo, yodo y astato. Estos elementos son demasiado reactivos para aparecer libremente en la naturaleza, pero sus compuestos están ampliamente distribuidos. Los cloruros son los más abundantes; aunque los fluoruros, bromuros y yoduros son menos comunes, están razonablemente disponibles. En esta sección, examinaremos la incidencia, la preparación y las propiedades de los halógenos. A continuación, examinaremos los compuestos halógenos con los metales representativos y, después, los interhalógenos. Esta sección concluirá con algunas aplicaciones de los halógenos.
### Incidencia y preparación
Todos los halógenos se encuentran en el agua de mar en forma de iones de haluro. La concentración del ion de cloruro es de 0,54 M; la de los demás haluros es inferior a 10-4 M. El flúor también se encuentra en minerales como el CaF2, el Ca(PO4)3F y el Na3AlF6. El cloruro también se encuentra en el Gran Lago Salado y el Mar Muerto, y en extensos lechos de sal que contienen NaCl, KCl o MgCl2. Parte del cloro de su cuerpo está presente en forma de ácido clorhídrico, que es un componente del ácido del estómago. Los compuestos de bromo se encuentran en el Mar Muerto y en las salmueras subterráneas. Los compuestos de yodo se encuentran en pequeñas cantidades en el salitre de Chile, las salmueras subterráneas y las algas marinas. El yodo es esencial para el funcionamiento de la glándula tiroides.
Las mejores fuentes de halógenos (excepto el yodo) son las sales de haluro. Es posible oxidar los iones de haluro a moléculas halógenas diatómicas libres por varios métodos, dependiendo de la facilidad de oxidación del ion de haluro. El flúor es el más difícil de oxidar, mientras que el yoduro es el más fácil.
El principal método para preparar el flúor es la oxidación electrolítica. El procedimiento de electrólisis más común es utilizar una mezcla fundida de fluoruro de hidrógeno de potasio, KHF2, y fluoruro de hidrógeno anhidro. La electrólisis provoca la descomposición del HF, formando flúor gaseoso en el ánodo e hidrógeno en el cátodo. Es necesario mantener los dos gases separados para evitar su recombinación explosiva para reformar el fluoruro de hidrógeno.
La mayor parte del cloro comercial procede de la electrólisis del ion de cloruro en soluciones acuosas de cloruro de sodio; se trata del proceso cloro-álcali comentado anteriormente. El cloro también es un producto de la producción electrolítica de metales como el sodio, el calcio y el magnesio a partir de sus cloruros fundidos. También es posible preparar cloro mediante la oxidación química del ion de cloruro en solución ácida con agentes oxidantes fuertes como el dióxido de manganeso (MnO2) o el dicromato de sodio (Na2Cr2O7). La reacción con el dióxido de manganeso es:
La preparación comercial del bromo implica la oxidación del ion de bromuro por el cloro:
El cloro es un agente oxidante más fuerte que el bromo. Este método es importante para la producción de casi todo el bromo doméstico.
Parte del yodo procede de la oxidación del cloruro de yodo, ICl, o del ácido yódico, HlO3. La preparación comercial del yodo utiliza la reducción del yodato de sodio, NaIO3, una impureza en los depósitos de salitre de Chile, con hidrógeno sulfito de sodio:
### Propiedades de los halógenos
El flúor es un gas amarillo pálido, el cloro es un gas amarillo verdoso, el bromo es un líquido marrón rojizo intenso y el yodo es un sólido cristalino negro grisáceo. El bromo líquido tiene una alta presión de vapor, y el vapor rojizo es fácilmente visible en la . Los cristales de yodo tienen una notable presión de vapor. Cuando se calientan suavemente, estos cristales se subliman y forman un hermoso vapor de color violeta intenso.
El bromo es solo ligeramente soluble en agua, pero es miscible en todas las proporciones en disolventes menos polares (o no polares) como el cloroformo, el tetracloruro de carbono y el disulfuro de carbono, formando soluciones que varían del amarillo al marrón rojizo, según la concentración.
El yodo es soluble en cloroformo, tetracloruro de carbono, disulfuro de carbono y muchos hidrocarburos, dando soluciones violetas de moléculas de I2. El yodo se disuelve muy poco en el agua, dando soluciones de color marrón. Es bastante soluble en soluciones acuosas de yoduros, con los que forma soluciones marrones. Estas soluciones marrones se deben a que las moléculas de yodo tienen orbitales d de valencia vacíos y pueden actuar como ácidos de Lewis débiles hacia el ion de yoduro. La ecuación para la reacción reversible del yodo (ácido de Lewis) con el ion de yoduro (base de Lewis) para formar el ion de triyoduro, es:
Cuanto más fácil sea la oxidación del ion de haluro, más difícil será que el halógeno actúe como agente oxidante. El flúor generalmente oxida un elemento hasta su estado de oxidación más alto, mientras que los halógenos más pesados no pueden hacerlo. Por ejemplo, cuando el exceso de flúor reacciona con el azufre, se forma el SF6. El cloro produce SCl2 y el bromo, S2Br2. El yodo no reacciona con el azufre.
El flúor es el agente oxidante más potente de los elementos conocidos. Oxida espontáneamente la mayoría de los otros elementos; por lo tanto, la reacción inversa, la oxidación de los fluoruros, es muy difícil de realizar. El flúor reacciona directamente y forma fluoruros binarios con todos los elementos excepto los gases nobles más ligeros (He, Ne y Ar). El flúor es un agente oxidante tan fuerte que muchas sustancias se inflaman al entrar en contacto con él. Las gotas de agua se inflaman en flúor y forman O2, OF2, H2O2, O3 y HF. La madera y el amianto se inflaman y arden con el flúor gaseoso. La mayoría de los metales calientes arden vigorosamente en el flúor. Sin embargo, es posible manipular el flúor en recipientes de cobre, hierro o níquel porque una película adherente de la sal de flúor pasiva sus superficies. El flúor es el único elemento que reacciona directamente con el gas noble xenón.
Aunque es un fuerte agente oxidante, el cloro es menos activo que el flúor. La mezcla de cloro e hidrógeno en la oscuridad hace que la reacción entre ellos sea imperceptiblemente lenta. La exposición de la mezcla a la luz hace que ambos reaccionen de forma explosiva. El cloro también es menos activo frente a los metales que el flúor, y las reacciones de oxidación suelen requerir temperaturas más altas. El sodio fundido se enciende en el cloro. El cloro ataca a la mayoría de los no metales (el C, el N2 y el O2 son notables excepciones), formando compuestos moleculares covalentes. El cloro suele reaccionar con los compuestos que solo contienen carbono e hidrógeno (hidrocarburos) añadiéndose a múltiples enlaces o por sustitución.
En el agua fría, el cloro sufre una reacción de desproporción:
La mitad de los átomos de cloro se oxidan al estado de oxidación 1+ (ácido hipocloroso), y la otra mitad se reduce al estado de oxidación 1- (ion de cloruro). Esta desproporción es incompleta, por lo que el agua clorada es una mezcla en equilibrio de moléculas de cloro, moléculas de ácido hipocloroso, iones de hidronio e iones de cloruro. Cuando se expone a la luz, esta solución sufre una descomposición fotoquímica:
El cloro no metálico es más electronegativo que cualquier otro elemento, excepto el flúor, el oxígeno y el nitrógeno. En general, los elementos muy electronegativos son buenos agentes oxidantes; por lo tanto, esperaríamos que el cloro elemental oxide todos los demás elementos excepto estos tres (y los gases nobles no reactivos). Su propiedad oxidante, de hecho, es la responsable de su uso principal. Por ejemplo, el cloruro de fósforo(V), un importante producto intermedio en la preparación de insecticidas y armas químicas, se fabrica oxidando el fósforo con cloro:
También se utiliza una gran cantidad de cloro para oxidar, y por tanto destruir, materiales orgánicos o biológicos en la purificación del agua y en el blanqueo.
Las propiedades químicas del bromo son similares a las del cloro, aunque el bromo es el agente oxidante más débil y su reactividad es menor que la del cloro.
El yodo es el menos reactivo de los halógenos. Es el agente oxidante más débil, y el ion de yoduro es el ion de haluro más fácilmente oxidable. El yodo reacciona con los metales, pero suele ser necesario calentarlo. No oxida otros iones de haluro.
En comparación con los demás halógenos, el yodo solo reacciona ligeramente con el agua. Las trazas de yodo en el agua reaccionan con una mezcla de almidón e ion de yoduro, formando un color azul intenso. Esta reacción es una prueba muy sensible para detectar la presencia de yodo en el agua.
### Haluros de los metales representativos
Se han preparado miles de sales de los metales representativos. Los haluros binarios son una importante subclase de sales. Una sal es un compuesto iónico formado por cationes y aniones, distintos de los iones de hidróxido u óxido. En general, es posible preparar estas sales a partir de los metales o de óxidos, hidróxidos o carbonatos. Ilustraremos los tipos generales de reacciones para preparar sales a través de las reacciones utilizadas para preparar haluros binarios.
Los compuestos binarios de un metal con los halógenos son los haluros. La mayoría de los haluros binarios son iónicos. Sin embargo, el mercurio, los elementos del grupo 13 con estados de oxidación 3+, el estaño(IV) y el plomo(IV) forman haluros binarios covalentes.
La reacción directa de un metal y un halógeno produce el haluro del metal. Algunos ejemplos de estas reacciones de reducción-oxidación son:
Si un metal puede presentar dos estados de oxidación, puede ser necesario controlar la estequiometría para obtener el haluro con el estado de oxidación más bajo. Por ejemplo, la preparación del cloruro de estaño(II) requiere una relación 1:1 de Sn a Cl2, mientras que la preparación del cloruro de estaño(IV) requiere una relación 1:2:
Los metales representativos activos (los que son más fáciles de oxidar que el hidrógeno) reaccionan con los haluros de hidrógeno gaseosos para producir haluros metálicos e hidrógeno. La reacción del zinc con el fluoruro de hidrógeno es:
Los metales representativos activos también reaccionan con soluciones de haluros de hidrógeno para formar hidrógeno y soluciones de los haluros correspondientes. Ejemplos de estas reacciones son:
Los hidróxidos, los carbonatos y algunos óxidos reaccionan con las soluciones de los haluros de hidrógeno para formar soluciones de sales de haluro. Es posible preparar otras sales mediante la reacción de estos hidróxidos, carbonatos y óxidos con soluciones acuosas de otros ácidos:
Algunos haluros y muchas de las otras sales de los metales representativos son insolubles. Es posible preparar estas sales solubles mediante reacciones de metátesis que se producen al mezclar soluciones de sales solubles (vea la ). Las reacciones de metátesis se examinan en el capítulo sobre la estequiometría de las reacciones químicas.
Varios haluros se encuentran en grandes cantidades en la naturaleza. El océano y las salmueras subterráneas contienen muchos haluros. Por ejemplo, el cloruro de magnesio en el océano es la fuente de los iones de magnesio utilizados en la producción de magnesio. Grandes depósitos subterráneos de cloruro de sodio, como la mina de sal que se muestra en la , se dan en muchas partes del mundo. Estos depósitos sirven como fuente de sodio y cloro en casi todos los demás compuestos que contienen estos elementos. El proceso cloro-álcali es un ejemplo.
### Interhalógenos
Los compuestos formados por dos o más halógenos diferentes son interhalógenos. Las moléculas interhalógenas están formadas por un átomo del halógeno más pesado unido por enlaces simples a un número impar de átomos del halógeno más ligero. Las estructuras de IF3, IF5 e IF7 se ilustran en la . Las fórmulas de otros interhalógenos, cada una de las cuales procede de la reacción de los respectivos halógenos, están en la .
Observe en la que el flúor es capaz de oxidar el yodo hasta su máximo estado de oxidación, 7+, mientras que el bromo y el cloro, que son más difíciles de oxidar, solo alcanzan el estado de oxidación 5+. El estado de oxidación 7+ es el límite para los halógenos. Dado que los halógenos más pequeños se agrupan en torno a uno más grande, el número máximo de átomos más pequeños posibles aumenta a medida que aumenta el radio del átomo más grande. Muchos de estos compuestos son inestables y la mayoría son extremadamente reactivos. Los interhalógenos reaccionan como los haluros que los componen; los fluoruros halógenos, por ejemplo, son agentes oxidantes más fuertes que los cloruros halógenos.
Los polihaluros iónicos de los metales alcalinos, como KI3, KICl2, KICl4, CsIBr2 y CsBrCl2, que contienen un anión compuesto por al menos tres átomos de halógeno, están estrechamente relacionados con los interhalógenos. Como se ha visto anteriormente, la formación del anión polihaluro es responsable de la solubilidad del yodo en soluciones acuosas que contienen un ion de yoduro.
### Aplicaciones
El ion de flúor y los compuestos de flúor tienen muchos usos importantes. Los compuestos de carbono, hidrógeno y flúor están sustituyendo a los freones (compuestos de carbono, cloro y flúor) como refrigerantes. El teflón es un polímero compuesto por unidades de CF2CF2. El ion de fluoruro se añade a los suministros de agua y a algunos dentífricos como SnF2 o NaF para combatir la caries. El flúor convierte parcialmente los dientes de Ca5(PO4)3(OH) en Ca5(PO4)3F.
El cloro es importante para blanquear la pasta de madera y la tela de algodón. El cloro reacciona con el agua para formar ácido hipocloroso, que oxida las sustancias coloreadas y las convierte en incoloras. Grandes cantidades de cloro son importantes en la cloración de hidrocarburos (sustituyendo el hidrógeno por el cloro) para producir compuestos como el tetracloruro (CCl4), el cloroformo (CHCl3) y el cloruro de etilo (C2H5Cl), y en la producción de policloruro de vinilo (Polyvinyl Chloride, PVC) y otros polímeros. El cloro también es importante para eliminar las bacterias en los suministros de agua de la comunidad.
El bromo es importante en la producción de ciertos tintes, y los bromuros de sodio y potasio se utilizan como sedantes. En una época, el bromuro de plata sensible a la luz era un componente de la película fotográfica.
El yodo en solución alcohólica con yoduro de potasio es un antiséptico (tintura de yodo). Las sales de yodo son esenciales para el buen funcionamiento de la glándula tiroidea; una carencia de yodo puede provocar el desarrollo de un bocio. La sal de mesa yodada contiene un 0,023% de yoduro de potasio. El yoduro de plata es útil en la siembra de nubes para inducir la lluvia; fue importante en la producción de películas fotográficas y el yodoformo, CHI3, es un antiséptico.
### Conceptos clave y resumen
Los halógenos forman haluros con elementos menos electronegativos. Los haluros de los metales varían de iónicos a covalentes; los haluros de los no metales son covalentes. Los interhalógenos se forman por la combinación de dos o más halógenos diferentes.
Todos los metales representativos reaccionan directamente con los halógenos elementales o con soluciones de los ácidos hidrohalicos (HF, HCl, HBr y HI) para producir haluros metálicos representativos. Otras preparaciones de laboratorio consisten en la adición de ácidos hidrohalicos acuosos a compuestos que contienen dichos aniones básicos, como hidróxidos, óxidos o carbonatos.
### Ejercicios de química del final del capítulo
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# Metales representativos, metaloides y no metales
## Incidencia, preparación y propiedades de los gases nobles
Los elementos del grupo 18 son los gases nobles (helio, neón, argón, criptón, xenón y radón). Se ganaron el nombre de "nobles" porque se suponía que no eran reactivos, ya que tenían capas de valencia llenas. En 1962, el Dr. Neil Bartlett, de la Universidad de Columbia Británica, demostró que esta suposición era falsa.
Estos elementos están presentes en la atmósfera en pequeñas cantidades. Algunos gases naturales contienen un 1 a 2 % de helio en masa. El helio se aísla del gas natural licuando los componentes condensables, dejando solo el helio como gas. Los Estados Unidos posee la mayor parte del suministro comercial mundial de este elemento en sus yacimientos de gas con helio. El argón, el neón, el criptón y el xenón proceden de la destilación fraccionada del aire líquido. El radón procede de otros elementos radiactivos. Más recientemente, se ha observado que este gas radiactivo está presente en cantidades muy pequeñas en suelos y minerales. Sin embargo, su acumulación en edificios bien aislados y herméticamente cerrados constituye un peligro para la salud, principalmente el cáncer de pulmón.
Los puntos de ebullición y de fusión de los gases nobles son extremadamente bajos en relación con los de otras sustancias de masas atómicas o moleculares comparables. Esto se debe a que solo están presentes las débiles fuerzas de dispersión de London, y estas fuerzas solo pueden mantener unidos a los átomos cuando el movimiento molecular es muy ligero, como ocurre a temperaturas muy bajas. El helio es la única sustancia conocida que no se solidifica al enfriarse a presión normal. Permanece en estado líquido cerca del cero absoluto (0,001 K) a presiones ordinarias, pero se solidifica a presiones elevadas.
El helio se utiliza para llenar globos y naves más ligeras que el aire porque no arde, lo que hace que su uso sea más seguro que el del hidrógeno. El helio a altas presiones no es un narcótico como el nitrógeno. Por lo tanto, las mezclas de oxígeno y helio son importantes para los buceadores que trabajan a altas presiones. El uso de una mezcla de helio y oxígeno evita el estado mental de desorientación conocido como narcosis de nitrógeno, el llamado rapto de las profundidades. El helio es importante como atmósfera inerte para la fusión y soldadura de metales fácilmente oxidables y para muchos procesos químicos sensibles al aire.
El helio líquido (punto de ebullición, 4,2 K) es un refrigerante importante para alcanzar las bajas temperaturas necesarias para la investigación criogénica, y es esencial para lograr las bajas temperaturas necesarias para producir la superconducción en los materiales superconductores tradicionales utilizados en potentes imanes y otros dispositivos. Esta capacidad de enfriamiento es necesaria para los imanes utilizados en las imágenes por resonancia magnética, un procedimiento de diagnóstico médico habitual. El otro refrigerante habitual es el nitrógeno líquido (punto de ebullición, 77 K), que es bastante más barato.
El neón es un componente de las lámparas y señales de neón. El paso de una chispa eléctrica a través de un tubo que contiene neón a baja presión genera el conocido brillo rojo del neón. Es posible cambiar el color de la luz mezclando vapor de argón o mercurio con el neón o utilizando tubos de vidrio de un color especial.
El argón era útil en la fabricación de bombillas eléctricas, ya que su menor conductividad térmica y su inercia química lo hacían preferible al nitrógeno para inhibir la vaporización del filamento de tungsteno y prolongar la vida útil de la bombilla. Los tubos fluorescentes suelen contener una mezcla de argón y vapor de mercurio. El argón es el tercer gas más abundante en el aire seco.
Los tubos de flash de criptón-xenón se utilizan para tomar fotografías de alta velocidad. Una descarga eléctrica a través de un tubo de este tipo da una luz muy intensa que solamente dura de un segundo. El criptón forma un difluoruro, KrF2, que es térmicamente inestable a temperatura ambiente.
Los compuestos estables de xenón se forman cuando el xenón reacciona con el flúor. El difluoruro de xenón, XeF2, se forma tras calentar un exceso de xenón gaseoso con flúor gaseoso y enfriarlo. El material forma cristales incoloros, que son estables a temperatura ambiente en una atmósfera seca. El tetrafluoruro de xenón, XeF4, y el hexafluoruro de xenón, XeF6, se preparan de forma análoga, con una cantidad estequiométrica de flúor y un exceso de flúor, respectivamente. Los compuestos con oxígeno se preparan sustituyendo los átomos de flúor de los fluoruros de xenón por oxígeno.
Cuando el XeF6 reacciona con el agua, se produce una solución de XeO3 y el xenón permanece en el estado de oxidación 6+:
El trióxido de xenón sólido y seco, XeO3, es extremadamente explosivo: detonará espontáneamente. Tanto el XeF6 como el XeO3 se desproporcionan en solución básica, produciendo xenón, oxígeno y sales del ion de perxenato, en el que el xenón alcanza su estado de oxidación máximo de 8+.
Al parecer, el radón forma RnF2; las pruebas de este compuesto proceden de las técnicas de rastreo radioquímico.
Los compuestos inestables de argón se forman a bajas temperaturas, pero no se conocen compuestos estables de helio y neón.
### Conceptos clave y resumen
La propiedad más significativa de los gases nobles (grupo 18) es su inactividad. Se presentan en bajas concentraciones en la atmósfera. Se utilizan como atmósferas inertes, señales de neón y como refrigerantes. Los tres gases nobles más pesados reaccionan con el flúor para formar fluoruros. Los fluoruros de xenón son los mejor caracterizados como materiales de partida para otros pocos compuestos de gases nobles.
### Ejercicios de química del final del capítulo
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# Metales de transición y química de coordinación
## Introducción
Tenemos contacto diario con muchos metales de transición. El hierro está presente en todas partes, desde las anillas de su cuaderno de espiral y los cubiertos de su cocina hasta los automóviles, los barcos, los edificios y la hemoglobina de su sangre. El titanio es útil en la fabricación de productos ligeros y duraderos, como marcos de bicicleta, caderas artificiales y joyas. El cromo es útil como revestimiento protector en las instalaciones de fontanería y en los detalles de automóviles.
Además de utilizarse en sus formas elementales puras, muchos compuestos que contienen metales de transición tienen otras numerosas aplicaciones. El nitrato de plata se utiliza para crear espejos, el silicato de circonio proporciona fricción en los frenos de los automóviles y muchos agentes importantes para combatir el cáncer, como el fármaco cisplatino y especies afines, son compuestos de platino.
La variedad de propiedades que presentan los metales de transición se debe a sus complejas capas de valencia. A diferencia de la mayoría de los metales del grupo principal, en los que normalmente se observa un solo estado de oxidación, la estructura de la capa de valencia de los metales de transición significa que suelen presentarse en varios estados de oxidación estables diferentes. Además, las transiciones de electrones en estos elementos pueden corresponderse con la absorción de fotones en el espectro electromagnético visible, dando lugar a compuestos coloreados. Debido a estos comportamientos, los metales de transición presentan una química rica y fascinante. |
# Metales de transición y química de coordinación
## Incidencia, preparación y propiedades de los metales de transición y sus compuestos
Los metales de transición se definen como aquellos elementos que tienen (o forman fácilmente) orbitales d parcialmente llenos. Como se muestra en la , los elementos del bloque de los grupos 3 a 11 son elementos de transición. Los elementos del bloque , también llamados metales de transición interna (los lantánidos y los actínidos), también cumplen este criterio porque el orbital d está parcialmente ocupado antes que los orbitales f. Los orbitales d se llenan con la familia del cobre (grupo 11); por esta razón, la siguiente familia (grupo 12) no son técnicamente elementos de transición. Sin embargo, los elementos del grupo 12 presentan algunas de las mismas propiedades químicas y suelen incluirse en los debates sobre los metales de transición. Algunos químicos tratan los elementos del grupo 12 como metales de transición.
Los elementos del bloque d se dividen en la primera serie de transición (los elementos Sc al Cu), la segunda serie de transición (los elementos Y al Ag) y la tercera serie de transición (el elemento La y los elementos Hf al Au). El actinio, Ac, es el primer miembro de la cuarta serie de transición, que incluye también el Rf y el Rg.
Los elementos del bloque f son los elementos Ce al Lu, que constituyen la serie de lantánidos (o serie de los lantanoides), y los elementos Th al Lr, que constituyen la serie de los actínidos (o serie de los actinoides). Como el lantano se comporta de forma muy parecida a los elementos lantánidos, se considera un elemento lantánido, aunque su configuración de electrones lo convierte en el primer miembro de la tercera serie de transición. Del mismo modo, el comportamiento del actinio hace que forme parte de la serie de los actínidos, aunque su configuración de electrones lo convierte en el primer miembro de la cuarta serie de transición.
Los elementos de transición tienen muchas propiedades en común con otros metales. Casi todos son sólidos duros y de alta fusión que conducen bien el calor y la electricidad. Forman fácilmente aleaciones y pierden electrones para formar cationes estables. Además, los metales de transición forman una gran variedad de compuestos de coordinación estables, en los que el átomo o ion metálico central actúa como un ácido de Lewis y acepta uno o más pares de electrones. Muchas moléculas e iones diferentes pueden donar pares solitarios al centro metálico, sirviendo como bases de Lewis. En este capítulo, nos centraremos principalmente en el comportamiento químico de los elementos de la primera serie de transición.
### Propiedades de los elementos de transición
Los metales de transición presentan una amplia gama de comportamientos químicos. Como se desprende de sus potenciales de reducción (vea el Apéndice H), algunos metales de transición son fuertes agentes reductores, mientras que otros tienen una reactividad muy baja. Por ejemplo, los lantánidos forman todos cationes acuosos estables 3+. La fuerza impulsora de estas oxidaciones es similar a la de los metales alcalinotérreos como el Be o el Mg, formando Be2+ y Mg2+. Por otro lado, materiales como el platino y el oro tienen potenciales de reducción mucho mayores. Su capacidad para resistir la oxidación los convierte en materiales útiles para la construcción de circuitos y joyas.
Los iones de los elementos más ligeros del bloque d, como el Cr3+, el Fe3+ y el Co2+, forman coloridos iones hidratados que son estables en el agua. Sin embargo, los iones del periodo inmediatamente inferior a estos (Mo3+, Ru3+ e Ir2+) son inestables y reaccionan fácilmente con el oxígeno del aire. La mayoría de los iones simples y estables en el agua formados por los elementos más pesados del bloque d son oxianiones como y
El rutenio, el osmio, el rodio, el iridio, el paladio y el platino son los metales del grupo del platino. Con dificultad, forman cationes simples que son estables en agua y, a diferencia de los elementos anteriores de la segunda y tercera serie de transición, no forman oxianiones estables.
Tanto los elementos del bloque d como los del bloque f reaccionan con los no metales para formar compuestos binarios; a menudo es necesario calentarlos. Estos elementos reaccionan con los halógenos para formar una variedad de haluros cuyo estado de oxidación va de 1+ a 6+. Al calentarse, el oxígeno reacciona con todos los elementos de transición excepto el paladio, el platino, la plata y el oro. Los óxidos de estos últimos metales pueden formarse con otros reactivos, pero se descomponen al calentarse. Los elementos del bloque f, los elementos del grupo 3 y los elementos de la primera serie de transición, excepto el cobre, reaccionan con soluciones acuosas de ácidos, formando gas hidrógeno y soluciones de las sales correspondientes.
Los metales de transición pueden formar compuestos con una amplia gama de estados de oxidación. Algunos de los estados de oxidación observados de los elementos de la primera serie de transición se muestran en la . A medida que nos desplazamos de izquierda a derecha por la primera serie de transición, vemos que el número de estados de oxidación comunes aumenta al principio hasta alcanzar un máximo hacia la mitad de la tabla, y luego disminuye. Los valores de la tabla son valores típicos; hay otros valores conocidos, y es posible sintetizar nuevas adiciones. Por ejemplo, en 2014, los investigadores lograron sintetizar un nuevo estado de oxidación del iridio (9+).
Para los elementos que van del escandio al manganeso (la primera mitad de la primera serie de transición), el estado de oxidación más alto corresponde a la pérdida de todos los electrones en los orbitales s y d de sus capas de valencia. El ion de titanio(IV), por ejemplo, se forma cuando el átomo de titanio pierde sus dos electrones 3d y dos 4s. Estos estados de oxidación más altos son las formas más estables del escandio, el titanio y el vanadio. Sin embargo, no es posible seguir eliminando todos los electrones de valencia de los metales a medida que se avanza en la serie. Se sabe que el hierro forma estados de oxidación de 2+ a 6+, siendo el hierro(II) y el hierro(III) los más comunes. La mayoría de los elementos de la primera serie de transición forman iones con una carga de 2+ o 3+ que son estables en el agua, aunque los de los primeros miembros de la serie pueden ser fácilmente oxidados por el aire.
Los elementos de la segunda y tercera serie de transición suelen ser más estables en estados de oxidación más altos que los elementos de la primera serie. En general, el radio atómico aumenta hacia abajo en un grupo, lo que hace que los iones de la segunda y tercera serie sean más grandes que los de la primera serie. Eliminar los electrones de los orbitales más alejados del núcleo es más fácil que eliminar los electrones cercanos al núcleo. Por ejemplo, el molibdeno y el wolframio, miembros del grupo 6, están limitados en su mayoría a un estado de oxidación de 6+ en solución acuosa. El cromo, el miembro más ligero del grupo, forma iones de Cr3+ estables en el agua y, en ausencia de aire, iones de Cr2+ menos estables. El sulfuro con el estado de oxidación más alto para el cromo es el Cr2S3, que contiene el ion de Cr3+. El molibdeno y el wolframio forman sulfuros en los que los metales presentan estados de oxidación 4+ y 6+.
### Preparación de los elementos de transición
Las antiguas civilizaciones conocían el hierro, el cobre, la plata y el oro. Los periodos de la historia de la humanidad conocidos como la Edad del Bronce y la Edad del Hierro marcan los avances en los que las sociedades aprendieron a aislar ciertos metales y a utilizarlos para fabricar herramientas y bienes. Los minerales naturales de cobre, plata y oro pueden contener altas concentraciones de estos metales en forma elemental (). El hierro, en cambio, se presenta en la tierra casi exclusivamente en formas oxidadas, como el óxido (Fe2O3). Los primeros utensilios de hierro conocidos se fabricaron con meteoritos de hierro. Los artefactos de hierro que se conservan y que datan de aproximadamente 4000 a 2500 a.C. son escasos, pero todos los ejemplos conocidos contienen aleaciones específicas de hierro y níquel que solo se dan en objetos extraterrestres, no en la Tierra. Tuvieron que pasar miles de años de avances tecnológicos para que las civilizaciones desarrollaran la fundición del hierro, la capacidad de extraer un elemento puro de sus minerales naturales y para que las herramientas de hierro se convirtieran en algo común.
Por lo general, los elementos de transición se extraen de minerales que se encuentran en una variedad de menas. Sin embargo, la facilidad de su recuperación varía mucho, dependiendo de la concentración del elemento en la mena, la identidad de los otros elementos presentes y la dificultad de reducir el elemento al metal libre.
En general, no es difícil reducir los iones de los elementos del bloque d al elemento libre. El carbono es un agente reductor suficientemente fuerte en la mayoría de los casos. Sin embargo, al igual que los iones de los metales más activos del grupo principal, los iones de los elementos del bloque f deben ser aislados por electrólisis o por reducción con un metal activo como el calcio.
Vamos a hablar de los procesos utilizados para el aislamiento del hierro, el cobre y la plata porque estos tres procesos ilustran los principales medios de aislamiento de la mayoría de los metales del bloque d. En general, cada uno de estos procesos implica tres etapas principales: tratamiento preliminar, fundición y refinado.
1. Tratamiento preliminar. En general, hay un tratamiento inicial de las menas para hacerlas aptas para la extracción de los metales. Esto suele implicar la trituración o molienda de la mena, la concentración de los componentes metálicos y, a veces, el tratamiento químico de estas sustancias para convertirlas en compuestos más fáciles de reducir al metal.
2. Fundición. El siguiente paso es la extracción del metal en estado fundido, un proceso llamado fundición, que incluye la reducción del compuesto metálico al metal. Las impurezas pueden eliminarse añadiendo un compuesto que forme una escoria, una sustancia con un punto de fusión bajo que puede separarse fácilmente del metal fundido.
3. Refinamiento. El último paso en la recuperación de un metal es refinarlo. Los metales de bajo punto de ebullición, como el zinc y el mercurio, pueden refinarse por destilación. Cuando se funden en una mesa inclinada, los metales de baja fusión, como el estaño, se alejan de las impurezas de mayor fusión. La electrólisis es otro método común para refinar metales.
### Aislamiento del hierro
La aplicación temprana del hierro a la fabricación de herramientas y armas fue posible gracias a la amplia distribución de los minerales de hierro y a la facilidad con la que los compuestos de hierro de las menas podían ser reducidos por el carbono. Durante mucho tiempo, el carbón vegetal fue la forma de carbón utilizada en el proceso de reducción. La producción y el uso del hierro se extendieron mucho más hacia 1620, cuando se introdujo el coque como agente reductor. El coque es una forma de carbón que se forma al calentar el carbón en ausencia de aire para eliminar las impurezas.
El primer paso en la metalurgia del hierro suele ser la tostación de la mena (calentamiento del mineral en el aire) para eliminar el agua, descomponer los carbonatos en óxidos y convertir los sulfuros en óxidos. A continuación, los óxidos se reducen en un alto horno de 80 a 100 pies de altura y unos 25 pies de diámetro () en el que se introducen continuamente el mineral tostado, el coque y la piedra caliza (CaCO3 impuro) en la parte superior. El hierro fundido y la escoria se retiran por la parte inferior. La totalidad de las existencias en un horno puede pesar varios cientos de toneladas.
Cerca de la parte inferior de un horno hay boquillas a través de las cuales se introduce aire precalentado en el horno. En cuanto entra el aire, el coque de la región de las boquillas se oxida a dióxido de carbono con la liberación de una gran cantidad de calor. El dióxido de carbono caliente pasa hacia arriba a través de la capa superpuesta de coque caliente, donde se reduce a monóxido de carbono:
El monóxido de carbono sirve como agente reductor en las regiones superiores del horno. Las reacciones individuales se indican en la .
Los óxidos de hierro se reducen en la región superior del horno. En la región central, la piedra caliza (carbonato de calcio) se descompone y el óxido de calcio resultante se combina con la sílice y los silicatos del mineral para formar la escoria. La escoria es mayoritariamente silicato de calcio y contiene la mayoría de los componentes comercialmente poco importantes del mineral:
Justo debajo de la mitad del horno, la temperatura es lo suficientemente alta como para fundir tanto el hierro como la escoria. Se acumulan en capas en el fondo del horno; la escoria menos densa flota sobre el hierro y lo protege de la oxidación. Varias veces al día, la escoria y el hierro fundido se retiran del horno. El hierro se traslada a las máquinas de fundición o a una planta siderúrgica ().
Gran parte del hierro producido se refina y se convierte en acero. El acero se fabrica a partir del hierro eliminando las impurezas y añadiendo sustancias como el manganeso, el cromo, el níquel, el tungsteno, el molibdeno y el vanadio para producir aleaciones con propiedades que hacen que el material sea adecuado para usos específicos. La mayoría de los aceros también contienen porcentajes pequeños pero definidos de carbono (0,04 % a 2,5 %). Sin embargo, en la fabricación del acero hay que eliminar gran parte del carbono que contiene el hierro; de lo contrario, el exceso de carbono haría que el hierro fuera frágil.
### Aislamiento del cobre
Los minerales de cobre más importantes contienen sulfuros de cobre (como la covellita, CuS), aunque a veces se encuentran óxidos de cobre (como la tenorita, CuO) e hidroxicarbonatos de cobre [como la malaquita, Cu2(OH)2CO3]. En la producción de cobre metálico, el mineral sulfurado concentrado se tuesta para eliminar parte del azufre en forma de dióxido de azufre. La mezcla restante, compuesta por Cu2S, FeS, FeO, y SiO2, se mezcla con piedra caliza, que sirve de fundente (un material que ayuda a eliminar las impurezas), y se calienta. La escoria fundida se forma a medida que el hierro y la sílice se eliminan mediante reacciones ácido-base de Lewis:
En estas reacciones, el dióxido de silicio se comporta como un ácido de Lewis, que acepta un par de electrones de la base de Lewis (el ion de óxido).
La reducción del Cu2S que queda después de la fundición se realiza soplando aire a través del material fundido. El aire convierte parte del Cu2S en Cu2O. Tan pronto como se forma el óxido de cobre(I), este es reducido por el sulfuro de cobre(I) restante a cobre metálico:
El cobre obtenido así se denomina cobre ampollado por su aspecto característico, que se debe a las ampollas de aire que contiene (). Este cobre impuro se funde en grandes placas, que se utilizan como ánodos en el refinado electrolítico del metal (que se describe en el capítulo sobre electroquímica).
### Aislamiento de la plata
La plata se presenta a veces en grandes pepitas (), pero con más frecuencia en vetas y depósitos relacionados. En una época, el bateo era un método eficaz para aislar las pepitas de plata y de oro. Debido a su baja reactividad, estos metales, y algunos otros, se presentan en depósitos en forma de pepitas. El descubrimiento del platino se debió a que los exploradores españoles en Centroamérica confundieron las pepitas de platino con la plata. Cuando el metal no está en forma de pepitas, suele ser útil emplear un proceso llamado hidrometalurgia para separar la plata de sus minerales. La hidrometalurgia consiste en separar un metal de una mezcla convirtiéndolo primero en iones solubles y luego extrayéndolo y reduciéndolo para precipitar el metal puro. En presencia de aire, los cánidos de metales alcalinos forman fácilmente el ion soluble de dicianoargentato(I), a partir de plata metálica o de compuestos que contienen plata, como el Ag2S y AgCl. Las ecuaciones representativas son:
La plata se precipita a partir de la solución de cianuro mediante la adición de iones de zinc o de hierro(II), que sirven de agente reductor:
### Compuestos de metales de transición
El enlace en los compuestos simples de los elementos de transición va de iónico a covalente. En sus estados de oxidación inferiores, los elementos de transición forman compuestos iónicos; en sus estados de oxidación superiores, forman compuestos covalentes o iones poliatómicos. La variación de los estados de oxidación que presentan los elementos de transición confiere a estos compuestos una química de reducción-oxidación basada en los metales. A continuación se describe la química de varias clases de compuestos que contienen elementos de la serie de transición.
### Haluros
Los haluros anhidros de cada uno de los elementos de transición pueden prepararse por reacción directa del metal con los halógenos. Por ejemplo:
El calentamiento de un haluro metálico con un metal adicional puede utilizarse para formar un haluro del metal con un estado de oxidación más bajo:
La estequiometría del haluro metálico que resulta de la reacción del metal con un halógeno está determinada por las cantidades relativas de metal y halógeno y por la fuerza del halógeno como agente oxidante. Por lo general, el flúor forma metales que contienen flúor en sus estados de oxidación más altos. Los demás halógenos no pueden formar compuestos análogos.
En general, la preparación de soluciones acuosas estables de los haluros de los metales de la primera serie de transición se realiza mediante la adición de un ácido hídrico a carbonatos, hidróxidos, óxidos u otros compuestos que contienen aniones básicos. Las reacciones de muestra son:
La mayoría de los metales de la primera serie de transición también se disuelven en ácidos, formando una solución de la sal y el hidrógeno gaseoso. Por ejemplo:
La polaridad de los enlaces con los metales de transición varía no solo en función de las electronegatividades de los átomos implicados, sino también del estado de oxidación del metal de transición. Recuerde que la polaridad de los enlaces es un espectro continuo en el que los electrones se comparten uniformemente (enlaces covalentes) en un extremo y los electrones se transfieren completamente (enlaces iónicos) en el otro. Ningún enlace es 100 % iónico, y el grado de distribución uniforme de los electrones determina muchas propiedades del compuesto. Los haluros de metales de transición con números de oxidación bajos forman más enlaces iónicos. Por ejemplo, el cloruro de titanio(II) y el cloruro de titanio(III) (TiCl2 y TiCl3) tienen puntos de fusión elevados que son característicos de los compuestos iónicos, pero el cloruro de titanio(IV) (TiCl4) es un líquido volátil, consistente con la existencia de enlaces covalentes titanio-cloro. Todos los haluros de los elementos más pesados del bloque d tienen características covalentes significativas.
El comportamiento covalente de los metales de transición con estados de oxidación más altos se ejemplifica con la reacción de los tetrahaluros metálicos con el agua. Al igual que el tetracloruro de silicio covalente, tanto los tetrahaluros de titanio como los de vanadio reaccionan con el agua para dar soluciones que contienen los correspondientes ácidos hidrohalicos y los óxidos metálicos:
### Óxidos
Al igual que con los haluros, la naturaleza del enlace en los óxidos de los elementos de transición viene determinada por el estado de oxidación del metal. Los óxidos con estados de oxidación bajos tienden a ser más iónicos, mientras que los que tienen estados de oxidación más altos son más covalentes. Estas variaciones en el enlace se deben a que las electronegatividades de los elementos no son valores fijos. La electronegatividad de un elemento aumenta con el incremento del estado de oxidación. Los metales de transición en estados de oxidación bajos tienen valores de electronegatividad más bajos que el oxígeno; por lo tanto, estos óxidos metálicos son iónicos. Los metales de transición en estados de oxidación muy altos tienen valores de electronegatividad cercanos a los del oxígeno, lo que hace que estos óxidos sean covalentes.
Los óxidos de la primera serie de transición pueden prepararse calentando los metales en aire. Estos óxidos son Sc2O3, TiO2, V2O5, Cr2O3, Mn3O4, Fe3O4, Co3O4, NiO, y CuO.
Alternativamente, estos óxidos y otros óxidos (con los metales en diferentes estados de oxidación) pueden producirse calentando los correspondientes hidróxidos, carbonatos u oxalatos en una atmósfera inerte. El óxido de hierro(II) puede prepararse calentando oxalato de hierro(II), y el óxido de cobalto(II) se produce calentando hidróxido de cobalto(II):
A excepción del CrO3 y el Mn2O7, los óxidos de metales de transición no son solubles en agua. Pueden reaccionar con ácidos y, en algunos casos, con bases. En general, los óxidos de los metales de transición con los estados de oxidación más bajos son básicos (y reaccionan con los ácidos), los intermedios son anfóteros y los estados de oxidación más altos son principalmente acídicos. Los óxidos metálicos básicos en un estado de oxidación bajo reaccionan con los ácidos acuosos para formar soluciones de sales y agua. Algunos ejemplos son la reacción del óxido de cobalto(II) que acepta protones del ácido nítrico y el óxido de escandio(III) que acepta protones del ácido clorhídrico:
Los óxidos de metales con estados de oxidación 4+ son anfóteros y la mayoría no son solubles ni en ácidos ni en bases. El óxido de vanadio(V), el óxido de cromo(VI) y el óxido de manganeso(VII) son acídicos. Reaccionan con soluciones de hidróxidos para formar sales de los oxianiones y Por ejemplo, la ecuación iónica completa para la reacción del óxido de cromo(VI) con una base fuerte está dada por:
El óxido de cromo(VI) y el óxido de manganeso(VII) reaccionan con el agua para formar los ácidos H2CrO4 y HMnO4, respectivamente.
### Hidróxidos
Cuando se añade un hidróxido soluble a una solución acuosa de una sal de un metal de transición de la primera serie de transición, se forma un precipitado gelatinoso. Por ejemplo, la adición de una solución de hidróxido de sodio a una solución de sulfato de cobalto produce un precipitado gelatinoso de color rosa o azul de hidróxido de cobalto(II). La ecuación iónica neta es:
En este y muchos otros casos, estos precipitados son hidróxidos que contienen el ion del metal de transición, iones de hidróxido y agua coordinada al metal de transición. En otros casos, los precipitados son óxidos hidratados compuestos por el ion metálico, los iones de óxido y el agua de hidratación:
Estas sustancias no contienen iones de hidróxido. Sin embargo, tanto los hidróxidos como los óxidos hidratados reaccionan con los ácidos para formar sales y agua. Al precipitar un metal de la solución, es necesario evitar un exceso de ion de hidróxido, ya que esto puede conducir a la formación del ion complejo, como se discute más adelante en este capítulo. Los hidróxidos metálicos precipitados pueden separarse para su posterior procesamiento o para la eliminación de residuos.
### Carbonatos
Muchos de los elementos de la primera serie de transición forman carbonatos insolubles. Es posible preparar estos carbonatos mediante la adición de una sal de carbonato soluble a una solución de una sal de metal de transición. Por ejemplo, el carbonato de níquel puede prepararse a partir de soluciones de nitrato de níquel y carbonato de sodio según la siguiente ecuación iónica neta:
Las reacciones de los carbonatos de metales de transición son similares a las de los carbonatos de metales activos. Reaccionan con los ácidos para formar sales metálicas, dióxido de carbono y agua. Al calentarse, se descomponen, formando los óxidos de metales de transición.
### Otras sales
En muchos aspectos, el comportamiento químico de los elementos de la primera serie de transición es muy similar al de los metales del grupo principal. En particular, los mismos tipos de reacciones que se utilizan para preparar sales de los principales metales del grupo pueden utilizarse para preparar sales iónicas simples de estos elementos.
A partir de metales más activos que el hidrógeno se pueden preparar diversas sales por reacción con los ácidos correspondientes: El escandio metálico reacciona con el ácido bromhídrico para formar una solución de bromuro de escandio:
Los compuestos comunes de los que acabamos de hablar también pueden utilizarse para preparar sales. Las reacciones implicadas incluyen las reacciones de óxidos, hidróxidos o carbonatos con ácidos. Por ejemplo:
Las reacciones de sustitución en las que intervienen sales solubles pueden utilizarse para preparar sales insolubles. Por ejemplo:
En nuestra discusión de los óxidos en esta sección, hemos visto que las reacciones de los óxidos covalentes de los elementos de transición con los hidróxidos forman sales que contienen oxianiones de los elementos de transición.
### Conceptos clave y resumen
Los metales de transición son elementos con orbitales d parcialmente llenos, situados en el bloque d de la tabla periódica. La reactividad de los elementos de transición varía mucho, desde metales muy activos, como el escandio y el hierro, hasta elementos casi inertes, como los metales del platino. El tipo de química utilizada en el aislamiento de los elementos a partir de sus minerales depende de la concentración del elemento en su mineral y de la dificultad de reducir los iones de los elementos a los metales. Los metales más activos son más difíciles de reducir.
Los metales de transición presentan un comportamiento químico típico de los metales. Por ejemplo, se oxidan en el aire al calentarse y reaccionan con los halógenos elementales para formar haluros. Los elementos que están por encima del hidrógeno en la serie de actividad reaccionan con los ácidos, produciendo sales y gas hidrógeno. Los óxidos, hidróxidos y carbonatos de compuestos de metales de transición en estados de oxidación bajos son básicos. Los haluros y otras sales son generalmente estables en el agua, aunque en algunos casos hay que excluir el oxígeno. La mayoría de los metales de transición forman una variedad de estados de oxidación estables, lo que les permite demostrar una amplia gama de reactividad química.
### Ejercicios de química del final del capítulo
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# Metales de transición y química de coordinación
## Química de coordinación de los metales de transición
La hemoglobina, la clorofila, la vitamina B-12 y el catalizador que se utiliza en la fabricación del polietileno contienen compuestos de coordinación. Los iones de los metales, especialmente los de transición, son susceptibles de formar complejos. Muchos de estos compuestos son muy coloridos (). En lo que queda de este capítulo, estudiaremos la estructura y el enlace de estos notables compuestos.
Recuerde que, en la mayoría de los compuestos de elementos del grupo principal, los electrones de valencia de los átomos aislados se combinan para formar enlaces químicos que satisfacen la regla del octeto. Por ejemplo, los cuatro electrones de valencia del carbono se superponen con los electrones de cuatro átomos de hidrógeno para formar el CH4. El electrón de valencia del sodio se suma a los siete electrones de valencia del cloro para formar la unidad de fórmula iónica NaCl (). Los metales de transición no se unen de esta manera. Forman principalmente enlaces covalentes de coordinación, una forma de interacción ácido-base de Lewis en la que un donante (base de Lewis) aporta los dos electrones del enlace a un aceptor de electrones (ácido de Lewis). El ácido de Lewis en los complejos de coordinación, a menudo llamado ion (o átomo) metálico central, es un metal de transición o un metal de transición interna, aunque los elementos del grupo principal también pueden formar compuestos de coordinación. Los donantes de base de Lewis, llamados ligandos, abarcan una gran variedad de sustancias químicas: átomos, moléculas o iones. El único requisito es que tengan uno o más pares de electrones, que puedan donarse al metal central. En la mayoría de los casos, se trata de un átomo donante con un par solitario de electrones que forma un enlace de coordinación con el metal.
La esfera de coordinación consta del ion o átomo metálico central más sus ligandos unidos. Los corchetes en una fórmula encierran la esfera de coordinación; los elementos fuera de los corchetes no forman parte de la esfera de coordinación. El número de coordinación del ion o átomo metálico central es el número de átomos donantes enlazados a este. El número de coordinación del ion de plata en [Ag(NH3)2]+ es dos (). Para el ion de cobre(II) en [CuCl4]2-, el número de coordinación es cuatro, mientras que para el ion de cobalto(II) en [Co(H2O)6]2+, el número de coordinación es seis. Cada uno de estos ligandos es monodentado, del griego "un diente", lo que significa que se conectan con el metal central a través de un solo átomo. En este caso, el número de ligandos y el número de coordinación son iguales.
Muchos otros ligandos se coordinan con el metal de forma más compleja. Los ligandos bidentados son aquellos en los que dos átomos se coordinan con el centro metálico. Por ejemplo, la etilendiamina (en, H2NCH2CH2NH2) contiene dos átomos de nitrógeno, cada uno de los cuales tiene un par solitario y sirve como base de Lewis (). Ambos átomos se coordinan con un solo centro metálico. En el complejo [Co(en)3]3+, hay tres ligandos bidentados en, y el número de coordinación del ion cobalto(III) es seis. Los números de coordinación más comunes son el dos, el cuatro y el seis, aunque se conocen ejemplos de todos los números de coordinación del 1 al 15.
Cualquier ligando que se una a un ion metálico central mediante más de un átomo donante es un ligando polidentado (o de "muchos dientes") porque puede morder el centro metálico con más de un enlace. El término quelato, del griego "garra", también se utiliza para describir este tipo de interacción. Muchos ligandos polidentados son ligandos quelantes; el complejo formado por uno o más de estos ligandos y un metal central es un quelato. El ligando quelante también se conoce como agente quelante. El ligando quelante sujeta el ion metálico como una pinza de cangrejo sujetaría una canica. La es un ejemplo de quelato. El complejo hemo de la hemoglobina es otro ejemplo importante (). Contiene un ligando polidentado con cuatro átomos donantes que se coordinan con el hierro.
Los ligandos polidentados se identifican a veces con prefijos que indican el número de átomos donantes en el ligando. Como hemos visto, los ligandos con un átomo donante, como NH3, Cl-, y H2O, son monodentados. Los ligandos con dos grupos donantes son bidentados. La etilendiamina, H2NCH2CH2NH2, y el anión del ácido glicina, () son ejemplos de ligandos bidentados. Los ligandos tridentados, tetradentados, pentadentados y hexadentados contienen tres, cuatro, cinco y seis átomos donantes, respectivamente. El ligando en hemo () es tetradentado.
### La designación de los complejos
La nomenclatura de los complejos se basa en un sistema que sugirió Alfred Werner, químico suizo y premio Nobel, cuyo extraordinario trabajo de hace más de 100 años sentó las bases para comprender mejor estos compuestos. Las siguientes cinco reglas se utilizan para designar los complejos:
1. Si un compuesto de coordinación es iónico, se nombra primero el catión y después el anión, de acuerdo con la nomenclatura habitual.
2. Se nombran primero los ligandos y luego el metal central. Se nombran los ligandos en orden alfabético. Los ligandos negativos (aniones) tienen nombres que se forman al añadir -o al nombre de la raíz del grupo. Para ver ejemplos, consulte la . En la mayoría de los ligandos neutros, se utiliza el nombre de la molécula. Las cuatro excepciones comunes son agua (H2O), amino (NH3), carbonilo (CO) y nitrosilo (NO). Por ejemplo, se designa [Pt(NH3)2Cl4] como diaminotetracloroplatino (IV).
3. Si hay más de un ligando de un tipo determinado, el número se indica con los prefijos di- (para dos), tri- (para tres), tetra- (para cuatro), penta- (para cinco) y hexa- (para seis). A veces, los prefijos bis- (para dos), tris- (para tres) y tetraquis- (para cuatro) se utilizan cuando el nombre del ligando ya incluye di-, tri- o tetra-, o cuando el nombre del ligando comienza con una vocal. Por ejemplo, el ion bis(bipiridilo)osmio(II) utiliza bis- para significar que hay dos ligandos unidos al Os, y cada ligando bipiridilo contiene dos grupos piridina (C5H4N).
Cuando el complejo es un catión o una molécula neutra, el nombre del átomo metálico central se escribe exactamente como el nombre del elemento y va seguido de un número romano entre paréntesis para indicar su estado de oxidación ( y ). Cuando el complejo es un anión, se añade el sufijo -ato a la raíz del nombre del metal, seguido de la designación en números romanos de su estado de oxidación (). A veces, se utiliza el nombre latino del metal cuando el nombre inglés es inapropiado. Por ejemplo, se utiliza ferrato en lugar de ironato, plumbato en lugar de leadato y estannato en lugar de tinato. El estado de oxidación del metal se determina en función de las cargas de cada ligando y de la carga global del compuesto de coordinación. Por ejemplo, en [Cr(H2O)4Cl2]Br, la esfera de coordinación (entre corchetes) tiene una carga de 1+ para equilibrar el ion de bromuro. Los ligandos de agua son neutros y los de cloruro son aniónicos con una carga de 1- cada uno. Para determinar el estado de oxidación del metal, fijamos la carga global igual a la suma de los ligandos y el metal: +1 = -2 + x, por lo que el estado de oxidación(x) es igual a 3+.
### Las estructuras de los complejos
Las estructuras más comunes de los complejos en los compuestos de coordinación son la octaédrica, la tetraédrica y la cuadrada plana (vea la ). En los complejos de metales de transición, el número de coordinación determina la geometría alrededor del ion metálico central. En la se comparan los números de coordinación con la geometría molecular:
A diferencia de los átomos del grupo principal, en los que tanto los electrones enlazantes como los no enlazantes determinan la forma molecular, los electrones d no enlazantes no cambian la disposición de los ligandos. Los complejos octaédricos tienen un número de coordinación de seis, y los seis átomos donantes están dispuestos en las esquinas de un octaedro alrededor del ion metálico central. Los ejemplos se muestran en la . Los aniones cloruro y nitrato en [Co(H2O)6]Cl2 y [Cr(en)3](NO3)3, y los cationes de potasio en K2[PtCl6], están fuera de los corchetes y no están unidos al ion metálico.
En los metales de transición con un número de coordinación de cuatro, son posibles dos geometrías diferentes: tetraédrica o cuadrada plana. A diferencia de los elementos del grupo principal, en los que estas geometrías se predicen a partir de la teoría VSEPR, se requiere un análisis más detallado de los orbitales de los metales de transición (que se trata en la sección sobre la Teoría del Campo Cristalino) para predecir qué complejos serán tetraédricos y cuáles serán cuadrangulares. En complejos tetraédricos como el [Zn(CN)4]2- (), cada uno de los pares de ligandos forma un ángulo de 109,5°. En los complejos cuadrados planos, como el [Pt(NH3)2Cl2], cada ligando tiene otros dos ligandos en ángulos de 90° (llamados posiciones cis) y otro ligando en un ángulo de 180°, en la posición trans.
### Isomerismo en los complejos
Los isómeros son especies químicas diferentes que tienen la misma fórmula química. Los complejos de metales de transición suelen existir como isómeros geométricos, en los que los mismos átomos están conectados a través de los mismos tipos de enlaces pero con diferencias en su orientación en el espacio. Los complejos de coordinación con dos ligandos diferentes en las posiciones cis y trans de un ligando de interés forman isómeros. Por ejemplo, el ion octaédrico [Co(NH3)4Cl2]+ tiene dos isómeros. En la configuración , los dos ligandos de cloruro son adyacentes entre sí (). El otro isómero, la configuración tiene los dos ligandos de cloruro uno frente al otro.
Los distintos isómeros geométricos de una sustancia son compuestos químicos diferentes. Presentan propiedades distintas, aunque tengan la misma fórmula. Por ejemplo, los dos isómeros de [Co(NH3)4Cl2]NO3 difieren en el color: la forma cis es violeta, y la forma trans es verde. Además, estos isómeros tienen momentos dipolares, solubilidades y reactividades diferentes. A modo de ejemplo de cómo la disposición en el espacio influye en las propiedades moleculares, consideremos la polaridad de los dos isómeros de [Co(NH3)4Cl2]NO3. Recuerde que la polaridad de una molécula o un ion viene determinada por los dipolos de enlace (que se deben a la diferencia de electronegatividad de los átomos enlazados) y su disposición en el espacio. En un isómero, los ligandos de cloruro cis provocan más densidad de electrones en un lado de la molécula que en el otro, lo que la hace polar. En el caso del isómero trans, cada ligando está directamente enfrente de un ligando idéntico, por lo que los dipolos de enlace se anulan y la molécula es apolar.
Otro tipo importante son los isómeros ópticos o enantiómeros, en los que dos objetos son imágenes especulares idénticas, pero no pueden alinearse para que todas las partes coincidan. Esto significa que los isómeros ópticos son imágenes especulares no superpuestas. Un ejemplo clásico son las manos, en el que la mano derecha y la izquierda son imágenes especulares la una de la otra, pero no se superponen. Los isómeros ópticos son muy importantes en la química orgánica y en la bioquímica porque los sistemas vivos suelen incorporar un determinado isómero óptico y no el otro. A diferencia de los isómeros geométricos, los pares de isómeros ópticos tienen propiedades idénticas (punto de ebullición, polaridad, solubilidad, etc.). Los isómeros ópticos apenas se diferencian en la forma en que inciden en la luz polarizada y en cómo reaccionan con otros isómeros ópticos. En el caso de los complejos de coordinación, muchos compuestos de coordinación como [M(en)3]n+ [en el cual Mn+ es un ion metálico central como el hierro(III) o el cobalto(II)] forman enantiómeros, como se muestra en la . Estos dos isómeros reaccionan de forma diferente con otros isómeros ópticos. Por ejemplo, las hélices de ADN son isómeros ópticos, y la forma que se da en la naturaleza (ADN diestro) se unirá solamente a un isómero de [M(en)3]n+ y no al otro.
El ion de [Co(en)2Cl2]+ presenta isomería geométrica (cis/trans), y su isómero cis existe como un par de isómeros ópticos ().
Los isómeros de enlace se producen cuando el compuesto de coordinación contiene un ligando que se une al centro del metal de transición a través de dos átomos diferentes. Por ejemplo, el ligando CN puede unirse a través del átomo de carbono (ciano) o a través del átomo de nitrógeno (isociano). Del mismo modo, el SCN- se une a través del átomo de azufre o de nitrógeno, para dar lugar a dos compuestos distintos ([Co(NH3)5SCN]2+ o [Co(NH3)5NCS]2+).
Los isómeros de ionización (o isómeros de coordinación) se producen cuando un ligando aniónico de la esfera de coordinación interna se sustituye por el contraión de la esfera de coordinación externa. Un simple ejemplo de dos isómeros de ionización son [CoCl6][Br] y [CoCl5Br][Cl].
### Complejos de coordinación en la naturaleza y la tecnología
La clorofila, el pigmento verde de las plantas, es un complejo que contiene magnesio (). Este es un ejemplo de un elemento del grupo principal en un complejo de coordinación. Las plantas parecen verdes porque la clorofila absorbe la luz roja y púrpura; la luz reflejada aparece, por tanto, verde. La energía resultante de la absorción de la luz se utiliza en la fotosíntesis.
Muchos otros complejos de coordinación también son de colores brillantes. El complejo cuadrado plano de cobre(II) azul de ftalocianina (de la ) es uno de los muchos que se utilizan como pigmentos o colorantes. Este complejo se utiliza en la tinta azul, los jeans y ciertas pinturas azules.
La estructura del hemo (), el complejo que contiene hierro en la hemoglobina, es muy similar a la de la clorofila. En la hemoglobina, el complejo rojo del hemo está unido a una gran molécula de proteína (globina) mediante la unión de la proteína al ligando del hemo. Las moléculas de oxígeno son transportadas por la hemoglobina en la sangre al estar unidas al centro de hierro. Cuando la hemoglobina pierde su oxígeno, el color cambia a un rojo azulado. La hemoglobina solo transporta oxígeno si el hierro es Fe2+; la oxidación del hierro a Fe3+ impide el transporte de oxígeno.
Los agentes complejantes se utilizan a menudo para ablandar el agua porque retienen iones como el Ca2+, Mg2+ y el Fe2+, que endurecen el agua. Muchos iones metálicos tampoco son deseables en los productos alimentarios porque pueden catalizar reacciones que cambian el color de los alimentos. Los complejos de coordinación sirven como conservantes. Por ejemplo, el ligando EDTA, (HO2CCH2)2NCH2CH2N(CH2CO2H)2, se coordina con los iones metálicos a través de seis átomos donantes e impide que los metales reaccionen (). Este ligando también se utiliza para secuestrar iones metálicos en la producción de papel, textiles y detergentes, y tiene usos farmacéuticos.
Los agentes complejantes que fijan los iones metálicos también se utilizan como fármacos. El dimercaprol (British anti-Lewisite, BAL), HSCH2CH(SH)CH2OH, es un medicamento desarrollado durante la Primera Guerra Mundial como antídoto para la lewisita, un agente químico de guerra a base de arsénico. El BAL se utiliza actualmente para tratar la intoxicación por metales pesados, como el arsénico, el mercurio, el talio y el cromo. El fármaco es un ligando y funciona al hacer un quelato hidrosoluble del metal; los riñones eliminan este quelato metálico (). Otro ligando polidentado, la enterobactina, que se aísla de ciertas bacterias, se utiliza para formar complejos de hierro y, de este modo, controlar la grave acumulación de hierro que se da en pacientes que padecen hemopatías, como la anemia de Cooley, que requieren transfusiones frecuentes. A medida que la sangre transfundida se descompone, los procesos metabólicos habituales que eliminan el hierro se sobrecargan, y el exceso de hierro puede acumularse hasta niveles mortales. La enterobactina forma un complejo hidrosoluble con el exceso de hierro, y el organismo puede eliminar este complejo de forma segura.
Los ligandos también se utilizan en la industria de la galvanoplastia. Cuando los iones metálicos se reducen para producir revestimientos metálicos finos, los metales se agrupan para formar cúmulos y nanopartículas. Cuando se utilizan complejos de coordinación metálica, los ligandos mantienen los átomos metálicos aislados entre sí. Se ha comprobado que muchos metales presentan una superficie más lisa, uniforme, de mejor aspecto y más adherente cuando se recubren con un baño que contiene el metal como ion complejo. Por lo tanto, complejos como el [Ag(CN)2]- y el [Au(CN)2]- se utilizan ampliamente en la industria de la galvanoplastia.
En 1965, científicos de la Universidad Estatal de Michigan descubrieron que había un complejo de platino que inhibe la división celular en ciertos microorganismos. Trabajos posteriores demostraron que se trataba del complejo cis-diaminodicloroplatino(II), [Pt(NH3)2(Cl)2], y que el isómero trans no era eficaz. La inhibición de la división celular indicó que este compuesto cuadrangular podría ser un agente anticanceroso. En 1978, la Administración de Alimentos y Medicamentos de los EE. UU. aprobó este compuesto, conocido como cisplatino, en el tratamiento de ciertas formas de cáncer. Desde entonces, se han desarrollado muchos compuestos semejantes de platino para el tratamiento del cáncer. En todos los casos, se trata de los isómeros cis y nunca de los isómeros trans. La porción de diamino (NH3)2 se mantiene con otros grupos, al sustituir la porción de dicloro [(Cl)2]. Los fármacos más nuevos son el carboplatino, el oxaliplatino y el satraplatino.
### Conceptos clave y resumen
Los elementos de transición y los elementos del grupo principal forman compuestos de coordinación, o complejos, en los que un átomo o ion metálico central está unido a uno o más ligandos mediante enlaces covalentes coordinados. Los ligandos con más de un átomo donante se denominan ligandos polidentados y forman quelatos. Las geometrías comunes que se encuentran en los complejos son la tetraédrica y la cuadrada plana (ambas con un número de coordinación de cuatro) y la octaédrica (con un número de coordinación de seis). Las configuraciones cis y trans son posibles en algunos complejos octaédricos y cuadrangulares. Además de estos isómeros geométricos, en ciertos complejos octaédricos son posibles los isómeros ópticos (moléculas o iones que son imágenes especulares pero no superpuestas). Los complejos de coordinación tienen una gran variedad de usos, como el transporte de oxígeno en la sangre, la purificación del agua y la industria farmacéutica.
### Ejercicios de química del final del capítulo
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# Metales de transición y química de coordinación
## Propiedades espectroscópicas y magnéticas de los compuestos de coordinación
El comportamiento de los compuestos de coordinación no puede explicarse adecuadamente con las mismas teorías que se utilizan para la química de los elementos del grupo principal. Las geometrías observadas de los complejos de coordinación no concuerdan con los orbitales hibridados en el metal central que se solapan con los orbitales del ligando, como predeciría la teoría del enlace de valencia. Los colores observados indican que los orbitales d se encuentran a menudo en diferentes niveles de energía en lugar de ser todos degenerados, es decir, de igual energía, como los tres orbitales p. Para explicar las estabilidades, estructuras, colores y propiedades magnéticas de los complejos de metales de transición, se ha elaborado un modelo de enlace diferente. Al igual que la teoría del enlace de valencia explica muchos aspectos del enlace en la química del grupo principal, la teoría del campo cristalino sirve para comprender y predecir el comportamiento de los complejos de metales de transición.
### Teoría del campo cristalino
Para explicar el comportamiento observado de los complejos de metales de transición (por ejemplo, cómo surgen los colores), se ha elaborado un modelo que implica interacciones electrostáticas entre los electrones de los ligandos y los electrones de los orbitales d no hibridados del átomo metálico central. Este modelo electrostático es la teoría del campo cristalino (TCC). Nos permite comprender, interpretar y predecir los colores, el comportamiento magnético y algunas estructuras de los compuestos de coordinación de los metales de transición.
La TCC se enfoca en los electrones no enlazados en el ion metálico central de los complejos de coordinación y no en los enlaces metal-ligando. Al igual que la teoría del enlace de valencia, la TCC apenas cuenta una parte del comportamiento de los complejos. Sin embargo, ahonda en la parte que la teoría de los enlaces de valencia no aborda. En su forma pura, la TCC desestima cualquier enlace covalente entre los ligandos y los iones metálicos. Tanto el ligando como el metal se tratan como cargas puntuales infinitesimales.
Todos los electrones son negativos, por lo que los electrones donados por los ligandos repelerán los electrones del metal central. Consideremos el comportamiento de los electrones en los orbitales d no hibridados en un complejo octaédrico. Los cinco orbitales d consisten en regiones con forma de lóbulo y están dispuestos en el espacio, como se muestra en la . En un complejo octaédrico, los seis ligandos se coordinan a lo largo de los ejes.
En un ion metálico no complejo en fase gaseosa, los electrones se distribuyen entre los cinco orbitales d de acuerdo con la regla de Hund, ya que los orbitales tienen todos la misma energía. Sin embargo, cuando los ligandos se coordinan con un ion metálico, las energías de los orbitales d ya no son las mismas.
En los complejos octaédricos, los lóbulos de dos de los cinco orbitales d, los orbitales y apuntan hacia los ligandos (). Estos dos orbitales se denominan orbitales (el símbolo en realidad se refiere a la simetría de los orbitales, pero lo utilizaremos como un nombre conveniente para estos dos orbitales en un complejo octaédrico). Los otros tres orbitales, d, d y d, tienen lóbulos que apuntan entre los ligandos y se denominan orbitales (de nuevo, el símbolo se refiere realmente a la simetría de los orbitales). A medida que seis ligandos se acercan al ion metálico a lo largo de los ejes del octaedro, sus cargas puntuales repelen los electrones de los orbitales d del ion metálico. Sin embargo, las repulsiones entre los electrones de los orbitales e (los orbitales y ) y los ligandos son mayores que las repulsiones entre los electrones de los orbitales t2 (los orbitales d, d, y d) y los ligandos. Esto se debe a que los lóbulos de los orbitales e apuntan directamente a los ligandos, mientras que los lóbulos de los orbitales t2 apuntan entre ellos. Así, los electrones de los orbitales e del ion metálico en un complejo octaédrico tienen energías potenciales más altas que las de los electrones de los orbitales t2. La diferencia de energía puede representarse como se indica en la .
La diferencia de energía entre los orbitales e y los orbitales t2 recibe el nombre de desdoblamiento del campo cristalino y se simboliza Δ, donde oct significa octaédrico.
La magnitud de Δoctt depende de muchos factores, como la naturaleza de los seis ligandos situados alrededor del ion metálico central, la carga del metal y si el metal utiliza orbitales 3d, 4d, o 5d. Diferentes ligandos producen diferentes desdoblamientos del campo cristalino. El desdoblamiento creciente del campo cristalino que causan los ligandos se expresa en la serie espectroquímica, de la que aquí se ofrece una versión abreviada:
En esta serie, los ligandos de la izquierda provocan pequeños desdoblamientos del campo cristalino y son ligandos de campo débil, mientras que los de la derecha provocan desdoblamientos mayores y son ligandos de campo fuerte. Así, el valor de Δoct para un complejo octaédrico con ligandos yodados (I-) es mucho menor que el valor de Δoct para el mismo metal con ligandos cianurados (CN-).
Los electrones en los orbitales d siguen el principio de Aufbau (“construcción”), el cual establece que los orbitales se llenarán para dar la menor energía total, igual que en la química del grupo principal. Cuando dos electrones ocupan el mismo orbital, las cargas similares se repelen. La energía necesaria para emparejar dos electrones en un mismo orbital se denomina energía de apareamiento (P). Los electrones siempre ocuparán individualmente cada orbital en un conjunto degenerado antes de aparearse. P tiene una magnitud semejante a la de Δoct. Cuando los electrones llenan los orbitales d, las magnitudes relativas de Δoct y P determinan qué orbitales estarán ocupados.
En [Fe(CN)6]4-, el fuerte campo de seis ligandos de cianuro produce un gran Δoct. En estas condiciones, los electrones necesitan menos energía para aparearse que para ser estimulados a los orbitales e (Δoct > P). Los seis electrones 3d del ion Fe2+ se aparean en los tres orbitales t2 (). Los complejos en los que los electrones se aparean debido al gran desdoblamiento del campo cristalino se denominan complejos de bajo espín porque el número de electrones desapareados (espines) es mínimo.
En cambio, en [Fe(H2O)6]2+, el campo débil de las moléculas de agua produce apenas un pequeño desdoblamiento del campo cristalino (Δoct < P). Dado que se requiere menos energía para que los electrones ocupen los orbitales e que para aparearse, habrá un electrón en cada uno de los cinco orbitales 3d antes de que se produzca el apareamiento. En cuanto a los seis electrones d en el centro del hierro(II) en [Fe(H2O)6]2+, habrá un par de electrones y cuatro electrones desapareados (). Los complejos como el ion [Fe(H2O)6]2+, en los que los electrones están desapareados porque el desdoblamiento del campo cristalino no es lo suficientemente grande como para que se apareen, se denominan complejos de alto espín porque el número de electrones desapareados (espines) es máximo.
Un razonamiento similar muestra por qué el ion [Fe(CN)6]3- es un complejo de bajo espín con un solo electrón desapareado, mientras que los iones [Fe(H2O)6]3+ y [FeF6]3- son complejos de alto espín con cinco electrones desapareados.
La otra geometría común es la cuadrada plana. Es posible considerar una geometría cuadrada plana como una estructura octaédrica con un par de ligandos trans eliminados. Se supone que los ligandos eliminados están en el eje z. Esto cambia la distribución de los orbitales d, ya que los orbitales en o cerca del eje zse vuelven más estables, y los que están en o cerca de los ejes x o y se vuelven menos estables. Esto hace que los conjuntos octaédricos t2 y e se desdoblen y den lugar a un patrón más complicado, como se representa a continuación:
### Momentos magnéticos de moléculas e iones
Las pruebas experimentales de las mediciones magnéticas apoyan la teoría de los complejos de alto y bajo espín. Recuerde que las moléculas como el O2 que contienen electrones desapareados son paramagnéticas. Las sustancias paramagnéticas son atraídas a los campos magnéticos. Muchos complejos de metales de transición tienen electrones desapareados y, por ende, son paramagnéticos. Las moléculas como el N2 y los iones como Na+ y [Fe(CN)6]4-, que no contienen electrones desapareados son diamagnéticos. Las sustancias diamagnéticas tienen una ligera tendencia a ser repelidas por los campos magnéticos.
Cuando un electrón de un átomo o de un ion está desapareado, el momento magnético debido a su espín hace que todo el átomo o el ion sea paramagnético. El tamaño del momento magnético de un sistema que contiene electrones desapareados está relacionado directamente con el número de dichos electrones: cuanto mayor sea el número de electrones desapareados, mayor será el momento magnético. Por lo tanto, el momento magnético observado se utiliza para determinar el número de electrones desapareados que están presentes. El momento magnético medido del d6 [Fe(CN)6]4- de bajo espín confirma que el hierro es diamagnético, mientras que el d6 [Fe(H2O)6]2+ de alto espín tiene cuatro electrones desapareados con un momento magnético que confirma esta disposición.
### Colores de los complejos de metales de transición
Cuando los átomos o las moléculas absorben la luz a la frecuencia adecuada, sus electrones se estimulan a orbitales de mayor energía. En muchos átomos y moléculas del grupo principal, los fotones absorbidos se encuentran en el rango ultravioleta del espectro electromagnético, que no detecta el ojo humano. En el caso de los compuestos de coordinación, la diferencia de energía entre los orbitales d permite absorber fotones en el rango visible.
El ojo humano percibe una mezcla de todos los colores, en las proporciones presentes en la luz solar, como luz blanca. Los colores complementarios, los que están situados uno frente al otro en la paleta de colores, también se utilizan en la visión del color. El ojo percibe una mezcla de dos colores complementarios, en las proporciones adecuadas, como luz blanca. Asimismo, cuando falta un color en la luz blanca, el ojo ve su complemento. Por ejemplo, cuando la luz blanca absorbe los fotones rojos, los ojos ven el color verde. Cuando se eliminan los fotones violetas de la luz blanca, los ojos ven el amarillo limón. El color azul del ion [Cu(NH3)4]2+ se debe a que este absorbe la luz naranja y roja, y quedan los colores complementarios azul y verde ().
Pequeños cambios en las energías relativas de los orbitales entre los que transitan los electrones pueden provocar cambios drásticos en el color de la luz absorbida. Por consiguiente, los colores de los compuestos de coordinación dependen de muchos factores. Como se muestra en la , distintos iones metálicos acuosos pueden tener diferentes colores. Además, distintos estados de oxidación de un metal pueden producir diferentes colores, como se muestra en el caso de los complejos de vanadio en el siguiente enlace.
Los ligandos específicos coordinados al centro metálico también influyen en el color de los complejos de coordinación. Por ejemplo, el complejo de hierro(II) [Fe(H2O)6]SO4 luce de color azul-verde porque el complejo de alto espín absorbe los fotones en las longitudes de onda del rojo (). En cambio, el complejo de hierro(II) de bajo espín K4[Fe(CN)6] luce de color amarillo pálido porque absorbe fotones violetas de mayor energía.
En general, los ligandos de campo fuerte causan un gran desdoblamiento en las energías de los orbitales d del átomo metálico central (Δoct). Los compuestos de coordinación de metales de transición con estos ligandos son de color amarillo, naranja o rojo porque absorben la luz violeta o azul de mayor energía. Por su parte, los compuestos de coordinación de los metales de transición con ligandos de campo débil son de color verde azulado, azul o índigo porque absorben la luz amarilla, naranja o roja de menor energía.
Un compuesto de coordinación del ion de Cu+ tiene una configuración d10, y todos los orbitales e están llenos. Para estimular un electrón a un nivel superior, como el orbital 4p, se necesitan fotones de muy alta energía. Esta energía corresponde a longitudes de onda muy cortas en la región ultravioleta del espectro. No se absorbe la luz visible, por lo que el ojo no ve ningún cambio, y el compuesto luce blanco o incoloro. Una solución con [Cu(CN)2]-, por ejemplo, es incolora. Adicionalmente, los complejos octaédricos de Cu2+ tienen una vacante en los orbitales e, y los electrones pueden excitarse a este nivel. La longitud de onda (energía) de la luz absorbida corresponde a la parte visible del espectro, y los complejos de Cu2+ son casi siempre de color azul, azul-verde-violeta o amarillo (). Aunque la TCC describe con éxito muchas propiedades de los complejos de coordinación, se requieren explicaciones de los orbitales moleculares (más allá del alcance introductorio que se ofrece aquí) para comprender plenamente el comportamiento de los complejos de coordinación.
### Conceptos clave y resumen
La teoría del campo cristalino trata las interacciones entre los electrones del metal y los ligandos como un simple efecto electrostático. La presencia de los ligandos cerca del ion metálico cambia las energías de los orbitales d del metal en relación con sus energías en el ion libre. Tanto el color como las propiedades magnéticas de un complejo pueden atribuirse a este desdoblamiento del campo cristalino. La magnitud del desdoblamiento (Δoct) depende de la naturaleza de los ligandos unidos al metal. Los ligandos de campo fuerte producen un gran desdoblamiento y favorecen los complejos de bajo espín, en los que los orbitales t2 se llenan completamente antes de que ningún electrón ocupe los orbitales e. Los ligandos de campo débil favorecen la formación de complejos de alto espín. Los orbitales t2 y e están ocupados individualmente antes que cualquiera sea doblemente ocupado.
### Ejercicios de química del final del capítulo
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# Química nuclear
## Introducción
Las reacciones químicas que hemos considerado en los capítulos anteriores implican cambios en la estructura electrónica de las especies implicadas, es decir, la disposición de los electrones alrededor de los átomos, iones o moléculas. La estructura nuclear, es decir, el número de protones y neutrones en los núcleos de los átomos implicados, no cambia durante las reacciones químicas.
En este capítulo se introducirá el tema de la química nuclear, que comenzó con el descubrimiento de la radiactividad en 1896 por el físico francés Antoine Becquerel y que ha adquirido mayor importancia durante los siglos XX y XXI, al sentar las bases de diversas tecnologías relacionadas con la energía, la medicina, la geología y muchas otras áreas. |
# Química nuclear
## Estructura y estabilidad nuclear
La química nuclear es el estudio de las reacciones que implican cambios en la estructura nuclear. El capítulo sobre átomos, moléculas e iones introdujo la idea básica de la estructura nuclear, que el núcleo de un átomo está compuesto por protones y, con la excepción de los neutrones . Recordemos que el número de protones en el núcleo recibe el nombre de número atómico (Z) del elemento, y la suma del número de protones y el número de neutrones es el número de masa (A). Los átomos con el mismo número atómico, pero con diferente número de masa son isótopos del mismo elemento. Cuando nos referimos a un solo tipo de núcleo, utilizamos el término nucleido y lo identificamos con la notación donde X es el símbolo del elemento, A es el número de masa y Z es el número atómico (por ejemplo, A menudo se hace referencia a un nucleido con el nombre del elemento seguido de un guion y el número de masa. Por ejemplo, se llama "carbono-14".
Los protones y los neutrones, denominados conjuntamente nucleones, se agrupan estrechamente en un núcleo. Con un radio aproximado de 10-15 metros, un núcleo es bastante pequeño comparado con el radio de todo el átomo, que es de unos 10-10 metros. Los núcleos son extremadamente densos en comparación con la materia en bruto, con una media de 1,8 1014 gramos por centímetro cúbico. Por ejemplo, el agua tiene una densidad de 1 gramo por centímetro cúbico, y el iridio (uno de los elementos más densos de los que se tenga conocimiento) tiene una densidad de 22,6 g/cm3. Si la densidad de la Tierra fuera igual al promedio de densidad nuclear, su radio sería apenas de unos 200 metros (el verdadero radio de la Tierra es de aproximadamente 6,4 106 metros, 30.000 veces mayor). El demuestra lo inmensa que pueden ser las densidades nucleares en el mundo natural.
Para mantener los protones cargados positivamente juntos en el pequeño volumen de un núcleo se necesitan fuerzas de atracción muy fuertes, ya que los protones cargados positivamente se repelen fuertemente a distancias tan cortas. La fuerza de atracción que mantiene unido el núcleo es la fuerza nuclear fuerte. (La fuerza fuerte es una de las cuatro fuerzas fundamentales que se conocen. Las otras son: la fuerza electromagnética, la fuerza gravitatoria y la fuerza nuclear débil). Esta fuerza actúa entre protones, entre neutrones y entre protones y neutrones. Es muy diferente de la fuerza electrostática, la cual mantiene a los electrones con carga negativa alrededor de un núcleo con carga positiva (la atracción entre cargas opuestas). En distancias inferiores a 10-15 metros y dentro del núcleo, la fuerza nuclear fuerte es mucho más potente que la repulsión electrostática entre protones; en distancias mayores y fuera del núcleo, es esencialmente inexistente.
### Energía de enlace nuclear
Como ejemplo sencillo de la energía asociada a la fuerza nuclear fuerte, consideremos el átomo de helio compuesto por dos protones, dos neutrones y dos electrones. La masa total de estas seis partículas subatómicas se calcula así:
Sin embargo, las mediciones por espectrometría de masas revelan que la masa del átomo de es de 4,0026 u: menos que las masas combinadas de sus seis partículas subatómicas constituyentes. Esta diferencia entre las masas calculadas y las medidas experimentalmente se conoce como el defecto de masa del átomo. En el caso del helio, el defecto de masa indica una "pérdida" de 4,0331 u - 4,0026 u = 0,0305 u. La pérdida de masa que acompaña a la formación de un átomo a partir de protones, neutrones y electrones se debe a la conversión de esa masa en energía, que evoluciona al formarse el átomo. La energía de enlace nuclear es la que se produce cuando los nucleones de los átomos se unen; también es la energía necesaria para romper un núcleo en sus protones y neutrones constituyentes. En comparación con las energías de enlace químico, la energía de enlace nuclear es mucho mayor, como lo aprenderemos en esta sección. Por consiguiente, los cambios de energía asociados a las reacciones nucleares son mucho mayores que los de las reacciones químicas.
La conversión entre masa y energía se identifica mejor con la ecuación de equivalencia entre masa y energía, tal y como la enunció Albert Einstein:
donde E es la energía, m es la masa de la materia que se convierte y c es la velocidad de la luz en el vacío. Esta ecuación sirve para calcular la cantidad que resulta cuando la materia se convierte en energía. Con esta ecuación de equivalencia entre masa y energía, la energía de enlace nuclear de un núcleo puede calcularse a partir de su defecto de masa, como se demuestra en el . Se utilizan varias unidades para las energías de enlace nuclear, como los electronvoltios (eV) , en los que 1 eV equivale a la cantidad de energía necesaria para mover la carga de un electrón a través de una diferencia de potencial eléctrico de 1 voltio, por lo que 1 eV = 1,602 10-19 J.
Debido a que los cambios de energía para romper y formar enlaces son tan pequeños comparados con aquellos para romper o formar núcleos, los cambios de masa durante todas las reacciones químicas ordinarias son prácticamente indetectables. Tal como se describe en el capítulo sobre termoquímica, las reacciones químicas más energéticas presentan entalpías del orden de miles de kJ/mol, lo que equivale a diferencias de masa en el rango de los nanogramos (10-9 g). Por otro lado, las energías de enlace nuclear suelen ser del orden de miles de millones de kJ/mol, lo que corresponde a diferencias de masa en el rango de los miligramos (10-3 g).
### Estabilidad nuclear
Un núcleo es estable si no puede transformarse en otra configuración sin añadir energía desde el exterior. De los miles de nucleidos que existen, unos 250 son estables. Un gráfico del número de neutrones frente al número de protones para los núcleos estables revela que los isótopos estables se sitúan en una banda estrecha. Esta región se conoce como banda de estabilidad (también llamada cinturón, zona o valle de estabilidad). La línea recta en la representa los núcleos que tienen un cociente 1:1 de protones y neutrones (cociente n:p). Observe que los núcleos estables más ligeros, en general, tienen igual número de protones y neutrones. Por ejemplo, el nitrógeno-14 tiene siete protones y siete neutrones. Sin embargo, los núcleos estables más pesados tienen cada vez más neutrones que protones. Por ejemplo: el hierro-56 tiene 30 neutrones y 26 protones, un cociente n:p de 1,15, mientras que el nucleido estable plomo-207 tiene 125 neutrones y 82 protones, un cociente n:p igual a 1,52. Esto se debe a que los núcleos más grandes tienen más repulsiones protón-protón, y requieren un mayor número de neutrones para proporcionar fuerzas potentes de compensación, superar estas repulsiones electrostáticas y mantener el núcleo unido.
Los núcleos que están a la izquierda o a la derecha de la banda de estabilidad son inestables y presentan radiactividad. Se transforman espontáneamente (decaen) en otros núcleos que están en la banda de estabilidad o más cerca de esta. Estas reacciones de decaimiento nuclear convierten un isótopo inestable (o radioisótopo) en otro más estable. En las siguientes secciones de este capítulo hablaremos de la naturaleza y los productos de este decaimiento radiactivo.
Pueden hacerse varias observaciones sobre la relación entre la estabilidad de un núcleo y su estructura. Los núcleos con un número par de protones, neutrones o ambos tienen más probabilidades de ser estables (vea la ). Los núcleos con cierto número de nucleones, conocidos como números mágicos, son estables frente al decaimiento nuclear. Estos números de protones o neutrones (2, 8, 20, 28, 50, 82 y 126) conforman las capas completas del núcleo. Su concepto es similar al de las capas estables de electrones que se observan en los gases nobles. Los núcleos que tienen números mágicos de protones y neutrones, como y se denominan "magia doble" y son especialmente estables. Estas tendencias en la estabilidad nuclear se pueden racionalizar tras considerar un modelo mecánico cuántico de estados de energía nuclear análogo al utilizado para describir los estados electrónicos anteriormente en este libro de texto. Los detalles de este modelo quedan fuera del alcance de este capítulo.
La estabilidad relativa de un núcleo está relacionada con su energía de enlace por nucleón: la energía total de enlace para el núcleo, dividida entre el número de nucleones en el núcleo. Vimos en el que la energía de enlace para un núcleo de es de 28,4 MeV. La energía de enlace por nucleón para un núcleo de es, por lo tanto:
En el , aprendemos a calcular la energía de enlace por nucleón de un nucleido en la curva que se indica en la .
### Conceptos clave y resumen
Un núcleo atómico está formado por protones y neutrones, llamados conjuntamente nucleones. Aunque los protones se repelen entre sí, el núcleo se mantiene unido por una fuerza de corto alcance, pero muy potente, llamada fuerza nuclear fuerte. Un núcleo tiene menos masa que la masa total de los nucleones que lo componen. Esta masa “faltante" es el defecto de masa, que se ha convertido en la energía de enlace que mantiene unido el núcleo según la ecuación de equivalencia entre masa y energía de Einstein, E = mc2. De los muchos nucleidos que existen, apenas un escaso número es estable. Los nucleidos con un número par de protones o neutrones, o aquellos con un número mágico de nucleones, son especialmente propensos a ser estables. Estos nucleidos estables ocupan una banda estrecha de estabilidad en un gráfico de número de protones frente al número de neutrones. La energía de enlace por nucleón es mayor para los elementos con números de masa cercanos a 56; estos son los núcleos más estables.
### Ecuaciones clave
### Ejercicios de química del final del capítulo
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# Química nuclear
## Ecuaciones nucleares
Las modificaciones de los núcleos que dan lugar a cambios en sus números atómicos, números de masa o estados energéticos son reacciones nucleares. Para describir una reacción nuclear, utilizamos una ecuación que identifica los nucleidos que intervienen en la reacción, sus números de masa y números atómicos, y las demás partículas que intervienen en la reacción.
### Tipos de partículas en las reacciones nucleares
En las reacciones nucleares intervienen muchas entidades. Las más comunes son los protones, los neutrones, las partículas alfa, las partículas beta, los positrones y los rayos gama, como se muestra en la . Los protones que también se representan con el símbolo y los neutrones son los constituyentes de los núcleos atómicos, y se han descrito anteriormente. Las partículas alfa que también se representan con el símbolo son núcleos de helio de alta energía. Las partículas beta que también se representan con el símbolo son electrones de alta energía, y los rayos gama son fotones de radiación electromagnética con energía extremadamente elevada. Los positrones que también se representan con el símbolo son electrones con carga positiva ("antielectrones"). Los subíndices y superíndices son necesarios para equilibrar las ecuaciones nucleares, aunque son opcionales en otras circunstancias. Por ejemplo, una partícula alfa es un núcleo de helio (He) con una carga de +2 y un número de masa de 4, por lo que se simboliza Esto funciona porque, en general, la carga del ion no es importante en el equilibrio de las ecuaciones nucleares.
Observe que los positrones son exactamente como los electrones, salvo que tienen la carga opuesta. Son el ejemplo más común de antimateria: partículas con la misma masa, pero con el estado opuesto de otra propiedad (por ejemplo, la carga) que la materia ordinaria. Cuando la antimateria se encuentra con la materia ordinaria, ambas se aniquilan y su masa se convierte en energía en forma de rayos gama (γ) (y otras partículas subnucleares mucho más pequeñas, que están fuera del alcance de este capítulo), según la ecuación de equivalencia entre masa y energía E = mc2, vista en la sección anterior. Por ejemplo, cuando un positrón y un electrón colisionan, ambos se aniquilan y se crean dos fotones de rayos gama:
Como se ha visto en el capítulo dedicado a la luz y la radiación electromagnética, los rayos gama componen una radiación electromagnética de corta longitud de onda y alta energía y son (mucho) más energéticos que los más conocidos rayos X, que se comportan como partículas en el sentido de la dualidad onda-partícula. Los rayos gama son un tipo de radiación electromagnética de alta energía que se produce cuando un núcleo experimenta una transición de un estado energético superior a otro inferior, de forma similar a como se produce un fotón por una transición electrónica de un nivel energético superior a otro inferior. Debido a las diferencias energéticas mucho mayores entre las capas de energía nuclear, los rayos gama que emanan de un núcleo tienen energías millones de veces mayores que la radiación electromagnética que emana de las transiciones electrónicas.
### Equilibrio de las reacciones nucleares
La ecuación de reacción química balanceada refleja el hecho de que, durante la reacción química, se rompen y se forman enlaces y se reordenan los átomos, pero el número total de átomos de cada elemento se conserva y no cambia. Una ecuación de reacción nuclear balanceada indica que hay un reordenamiento durante una reacción nuclear, pero de nucleones (partículas subatómicas dentro de los núcleos de los átomos) en lugar de átomos. Las reacciones nucleares también siguen las leyes de conservación, y se equilibran de dos maneras:
1. La suma de los números de masa de los reactivos es igual a la suma de los números de masa de los productos.
2. La suma de las cargas de los reactivos es igual a la suma de las cargas de los productos.
Si se conoce el número atómico y el número de masa de todas las partículas de una reacción nuclear menos una, podemos identificar la partícula al equilibrar la reacción. Por ejemplo, podríamos determinar que es un producto de la reacción nuclear de y si supiéramos que un protón, era uno de los dos productos. La muestra cómo podemos identificar un nucleido al equilibrar la reacción nuclear.
A continuación, se presentan las ecuaciones de varias reacciones nucleares que tienen un papel importante en la historia de la química nuclear:
1. El primer elemento inestable de origen natural que se aisló, el polonio, fue descubierto por la científica polaca Marie Curie y su marido Pierre en 1898. Decae y emite partículas α:
2. El primer nucleido que se preparó por medios artificiales fue un isótopo del oxígeno, el 17O. Lo hizo Ernest Rutherford en 1919 mediante el bombardeo de átomos de nitrógeno con partículas α:
3. James Chadwick descubrió el neutrón en 1932, como una partícula neutra hasta entonces desconocida, producida junto con el 12C mediante la reacción nuclear entre 9Be y 4He:
4. El primer elemento que se preparó y que no se da de forma natural en la Tierra, el tecnecio, lo crearon mediante el bombardeo del molibdeno con deuterones (hidrógeno pesado, , Emilio Segre y Carlo Perrier en 1937:
5. La primera reacción nuclear en cadena controlada se llevó a cabo en un reactor de la Universidad de Chicago en 1942. Una de las muchas reacciones fue:
### Conceptos clave y resumen
Los núcleos pueden sufrir reacciones que cambian su número de protones, su número de neutrones o su estado energético. En las reacciones nucleares intervienen muchas partículas diferentes. Las más comunes son los protones, los neutrones, los positrones (electrones con carga positiva), las partículas alfa (α) (núcleos de helio de alta energía), las partículas beta (β) (electrones de alta energía) y los rayos gama (γ) (que componen la radiación electromagnética de alta energía). Al igual que las reacciones químicas, las reacciones nucleares están siempre balanceadas. Cuando se produce una reacción nuclear, ni la masa total (número) ni la carga total cambian.
### Ejercicios de química del final del capítulo
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# Química nuclear
## Decaimiento radiactivo
Tras el descubrimiento un tanto fortuito de la radiactividad por parte de Becquerel, muchos científicos destacados comenzaron a investigar este nuevo e intrigante fenómeno. Entre ellos se encuentran Marie Curie (la primera mujer en ganar un premio Nobel y la única persona que ha ganado dos premios Nobel en ciencias diferentes: química y física), quien fue la primera en acuñar el término "radiactividad", y Ernest Rutherford (de la fama del experimento de la lámina de oro), quien investigó y bautizó a tres de los tipos más comunes de radiación. A principios del siglo XX se descubrieron muchas sustancias radiactivas, se investigaron y cuantificaron las propiedades de la radiación y se formó una sólida comprensión de la radiación y el decaimiento nuclear.
El cambio espontáneo de un nucleido inestable en otro es el decaimiento radiactivo. El nucleido inestable se denomina nucleido padre; el nucleido resultante del decaimiento se conoce como nucleido hija. El nucleido hija puede ser estable o decaer. La radiación que se emite durante el decaimiento radiactivo es tal que el nucleido hija se encuentra más cerca de la banda de estabilidad que el nucleido padre, por lo que la ubicación de un nucleido con respecto a la banda de estabilidad sirve de guía para el tipo de decaimiento que sufrirá ().
### Tipos de decaimiento radiactivo
Los experimentos de Ernest Rutherford sobre la interacción de la radiación con un campo magnético o eléctrico () le permitieron determinar que un tipo de radiación consistía en partículas α cargadas positivamente y relativamente masivas; un segundo tipo estaba formado por partículas β cargadas negativamente y mucho menos masivas, y un tercero eran ondas electromagnéticas sin carga, los rayos γ. Ahora sabemos que las partículas α son núcleos de helio de alta energía, las partículas β son electrones de alta energía y la radiación γ compone la radiación electromagnética de alta energía. Clasificamos los diferentes tipos de decaimiento radiactivo según la radiación producida.
El decaimiento alfa (α) es la emisión de una partícula α desde el núcleo. Por ejemplo, el polonio 210 sufre un decaimiento α:
El decaimiento alfa se produce principalmente en los núcleos pesados (A > 200, Z > 83). Dado que la pérdida de una partícula α da lugar a un nucleido hija con un número de masa cuatro unidades menor y un número atómico dos unidades menor que los del nucleido padre, el nucleido hija tiene un cociente n:p mayor que el nucleido padre. Si el nucleido padre que sufre el decaimiento α se encuentra por debajo de la banda de estabilidad (consulte la ), el nucleido hija se encontrará más cerca de la banda.
El decaimiento beta (β) es la emisión de un electrón desde un núcleo. El yodo-131 es un ejemplo de nucleido que sufre un decaimiento β:
El decaimiento beta, que puede considerarse como la conversión de un neutrón en un protón y una partícula β, se observa en los nucleidos con un gran cociente n:p. La partícula beta (electrón) emitida procede del núcleo atómico y no es ninguno de los electrones que rodean el núcleo. Estos núcleos se encuentran por encima de la banda de estabilidad. La emisión de un electrón no cambia el número de masa del nucleido, pero sí aumenta el número de sus protones y disminuye el de sus neutrones. En consecuencia, el cociente n:p disminuye, y el nucleido hija se encuentra más cerca de la banda de estabilidad que el nucleido padre.
La emisión gama (emisión γ) se observa cuando un nucleido se forma en un estado estimulado y luego decae a su estado básico con la emisión de un rayo γ, un quantum de radiación electromagnética de alta energía. La presencia de un núcleo en estado excitado se indica con un asterisco (*). El cobalto 60 emite radiación γ y se utiliza en muchas aplicaciones, incluido el tratamiento del cáncer:
No hay ningún cambio en el número de masa ni en el número atómico durante la emisión de un rayo γ, a menos que la emisión γ acompañe a uno de los otros modos de decaimiento.
La emisión de positrones (decaimiento β+) es la de un positrón desde el núcleo. El oxígeno-15 es un ejemplo de nucleido que sufre la emisión de positrones:
La emisión de positrones se observa en los nucleidos en los que el cociente n:p es bajo. Estos nucleidos se encuentran por debajo de la banda de estabilidad. El decaimiento de positrones es la conversión de un protón en un neutrón con la emisión de un positrón. El cociente n:p aumenta, y el nucleido hija se encuentra más cerca de la banda de estabilidad que el nucleido padre.
La captura de electrones tiene lugar cuando el núcleo de un átomo captura uno de sus electrones internos. Por ejemplo, el potasio-40 sufre la captura de electrones:
La captura de electrones tiene lugar cuando un electrón de la capa interna se combina con un protón y se convierte en neutrón. La pérdida de un electrón de la capa interna deja una vacante que ocupará uno de los electrones externos. Al caer el electrón exterior en la vacante, emitirá energía. En la mayoría de los casos, la energía emitida será en forma de rayos X. Al igual que la emisión de positrones, la captura de electrones tiene lugar para los núcleos "ricos en protones" que se encuentran por debajo de la banda de estabilidad. La captura de electrones tiene el mismo efecto sobre el núcleo que la emisión de positrones: el número atómico disminuye en uno y el número de masa no cambia. Esto aumenta el cociente n:p, y el nucleido hija se encuentra más cerca de la banda de estabilidad que el nucleido padre. Difícilmente se puede predecir si hay captura de electrones o emisión de positrones. La elección se debe principalmente a factores cinéticos, por lo que sería el más probable aquel que requiere la menor energía de activación.
La resume estos tipos de decaimiento, junto con sus ecuaciones y cambios en los números atómicos y de masa.
### Serie de decaimiento radiactivo
Los isótopos radiactivos naturales de los elementos más pesados se dividen en cadenas de desintegraciones sucesivas, o decaimientos, y todas las especies de una cadena constituyen una familia radiactiva o serie de decaimiento radiactivo. Tres de estas series abarcan la mayoría de los elementos naturalmente radiactivos de la tabla periódica. Son las series del uranio, de los actínidos y del torio. La del neptunio sería la cuarta serie, que ya no es significativa en la Tierra debido a la corta semivida de las especies implicadas. Cada serie se caracteriza por un nucleido padre (primer miembro), con una semivida prolongada, y una serie de nucleidos hijas que, en última instancia, conducen a un producto final estable, es decir, un nucleido en la banda de estabilidad (). En las tres series, el producto final es un isótopo estable del plomo. La serie del neptunio, que anteriormente se creía que terminaba con el bismuto-209, termina con el talio-205.
### Semividas radiactivas
El decaimiento radiactivo sigue una cinética de primer orden. Dado que las reacciones de primer orden ya se han tratado en detalle en el capítulo de cinética, ahora aplicaremos esos conceptos a las reacciones de decaimiento nuclear. Cada nucleido radiactivo tiene una semivida (t1/2) constante y característica: el tiempo necesario para que la mitad de los átomos de una muestra decaiga. La semivida de un isótopo nos permite determinar cuánto tiempo estará disponible una muestra de un isótopo útil, y cuánto tiempo deberá almacenarse una muestra de un isótopo indeseable o peligroso antes de que decaiga hasta un nivel de radiación lo suficientemente bajo como para que deje de ser un problema.
Por ejemplo, el cobalto-60, un isótopo que emite rayos gama y que se utiliza para tratar el cáncer, tiene una semivida de 5,27 años (). En una determinada fuente de cobalto-60, dado que la mitad de la los núcleos decaen cada 5,27 años, tanto la cantidad de material como la intensidad de la radiación emitida se reducen a la mitad cada 5,27 años. (Observe que, para una sustancia determinada, la intensidad de la radiación que produce es directamente proporcional a la tasa de decaimiento y a la cantidad de la sustancia). Esto es lo que se prevé en un proceso que sigue una cinética de primer orden. Por lo tanto, una fuente de cobalto 60 que se utiliza para el tratamiento del cáncer deberá reemplazarse regularmente para seguir siendo eficaz.
Dado que el decaimiento nuclear sigue una cinética de primer orden, podemos adaptar las relaciones matemáticas que se utilizan en las reacciones químicas de primer orden. Generalmente sustituimos el número de núcleos, N, por la concentración. Si la tasa se expresa en decaimientos nucleares por segundo, nos referimos a esta como la actividad de la muestra radiactiva. La tasa de decaimiento radiactivo es:
tasa de decaimiento = λN con λ = la constante de decaimiento del radioisótopo en cuestión.
La constante de decaimiento, λ, que es la misma que la constante de velocidad que se analiza en el capítulo de cinética. Es posible expresar la constante de decaimiento en términos de semivida, t1/2:
Las ecuaciones de primer orden que relacionan cantidad, N y tiempo son:
donde N0 es el número inicial de núcleos o moles del isótopo, y N es el número de núcleos/moles que quedan en el tiempo t. La aplica estos cálculos para hallar las tasas de decaimiento radiactivo de determinados nucleidos.
Ya que cada nucleido tiene un número específico de nucleones, un equilibrio particular de repulsión y atracción, y su propio grado de estabilidad, las semividas de los nucleidos radiactivos varían mucho. Por ejemplo: la semivida de es 1,9 1019 años; es de 24.000 años; es de 3,82 días; y el elemento-111 (Rg de roentgenio) es de 1,5 10-3 segundos. Las semividas de una serie de isótopos radiactivos importantes para la medicina se muestran en la , y otros se enumeran en el Apéndice M.
### Datación radiométrica
Varios radioisótopos tienen semividas y otras propiedades que sirven para "datar" el origen de objetos como artefactos arqueológicos, antiguos organismos vivos o formaciones geológicas. Este proceso es la datación radiométrica y ha sido la causa de muchos descubrimientos científicos revolucionarios sobre la historia geológica de la Tierra, la evolución de la vida y la historia de la civilización humana. Exploraremos algunos de los tipos más comunes de datación radiactiva y cómo funcionan los isótopos particulares de cada tipo.
### Datación radiactiva con carbono 14
La radiactividad del carbono 14 proporciona un método para datar objetos que formaron parte de un organismo vivo. Este método de datación radiométrica, que también se denomina datación por radiocarbono o por carbono 14, es exacto para datar sustancias que contienen carbono de hasta unos 30.000 años de antigüedad, y puede proporcionar fechas bastante precisas, hasta un máximo de unos 50.000 años.
El carbono natural consta de tres isótopos: que constituye aproximadamente el 99 % del carbono de la Tierra; alrededor del 1 % del total, y trazas de El carbono 14 se forma en la atmósfera superior por la reacción de los átomos de nitrógeno con los neutrones de los rayos cósmicos en el espacio:
Todos los isótopos del carbono reaccionan con el oxígeno para producir moléculas de CO2. El cociente de a depende del cociente de a en la atmósfera. La abundancia natural de en la atmósfera es de aproximadamente 1 parte por trillón. Hasta ahora esto ha sido generalmente constante a lo largo del tiempo, como se evidencia en las muestras encontradas de gas atrapadas en el hielo. La incorporación de y en las plantas es una parte regular del proceso de fotosíntesis, lo que significa que el cociente que se encuentra en una planta viva es igual al cociente en la atmósfera. No obstante, cuando la planta muere, ya no atrapa carbono a través de la fotosíntesis. Dado que es un isótopo estable y no sufre decaimiento radiactivo, su concentración en la planta no cambia. Sin embargo, el carbono-14 decae por emisión β con una semivida de 5730 años:
Por lo tanto, el cociente disminuye gradualmente después de la muerte de la planta. La disminución del cociente con el tiempo aporta una medida del tiempo que ha transcurrido desde la muerte de la planta (o de otro organismo que se comió la planta). La representa visualmente este proceso.
Por ejemplo, con la semivida de en 5730 años, si el cociente en un objeto de madera encontrado en una excavación arqueológica es la mitad que en un árbol vivo, esto indica que el objeto de madera tiene 5730 años. Pueden obtenerse determinaciones muy exactas de cocientes a partir de muestras ínfimas (de hasta un miligramo) con un espectrómetro de masas.
Se han producido algunos cambios significativos y bien documentados en el cociente . La exactitud de la aplicación directa de esta técnica depende de que el cociente en una planta viva sea el mismo ahora que en una época anterior, pero esto no siempre es válido. Debido a la creciente acumulación de moléculas de CO2 (en gran parte en la atmósfera a causa de la combustión de combustibles fósiles (en la que esencialmente todo el ha decaído), el cociente de en la atmósfera puede estar cambiando. Este aumento por el hombre de en la atmósfera hace que disminuya el cociente , lo que a su vez afecta el cociente en los organismos vivos en la Tierra. Sin embargo, afortunadamente podemos utilizar otros datos, como la datación de los árboles mediante el examen de los anillos de crecimiento anual, para calcular los factores de corrección. Con estos factores de corrección, se pueden determinar las fechas exactas. En general, la datación radiactiva funciona apenas durante unas 10 semividas; en consecuencia, el límite de la datación por carbono 14 es de unos 57.000 años.
### Datación radiactiva con nucleidos distintos del carbono 14
En la datación radiactiva también se utilizan otros nucleidos radiactivos con semividas más prolongadas para datar acontecimientos más antiguos. Por ejemplo, el uranio-238 (que decae en una serie de pasos hasta convertirse en plomo-206) se utiliza para establecer la edad de las rocas (y la edad aproximada de las rocas más antiguas de la Tierra). Dado que el U-238 tiene una semivida de 4.500 millones de años, la mitad del U-238 original tarda ese tiempo en decaer en Pb-206. En una muestra de roca que no contiene cantidades apreciables de Pb-208, el isótopo más abundante del plomo, supondríamos que el plomo no estaba presente cuando se formó la roca. Por consiguiente, al medir y analizar el cociente de U-238:Pb-206, determinamos la edad de la roca. Esto supone que todo el plomo-206 presente proviene del decaimiento del uranio-238. Si hay presencia adicional de plomo-206, lo que se indica por la presencia de otros isótopos de plomo en la muestra, es necesario hacer un ajuste. En la datación por argón potásico se utiliza un método parecido. El K-40 decae por emisión de positrones y capta electrones para formar Ar-40 con una semivida de 1.250 millones de años. Si se tritura una muestra de roca y se mide la cantidad de gas Ar-40 que escapa, la determinación del cociente Ar-40:K-40 arroja la edad de la roca. Otros métodos, como la datación por rubidio-estroncio (el Rb-87 decae en Sr-87, con una semivida de 48.800 millones de años), funcionan según el mismo principio. Para calcular el límite inferior de la edad de la Tierra, los científicos determinan la edad de diversas rocas y minerales, partiendo del supuesto de que la Tierra es más antigua que las rocas y minerales más antiguos de su corteza. Desde 2014, las rocas más antiguas que se conocen en la Tierra son los circones de Jack Hills, en Australia, que, según la datación con uranio y plomo, tienen casi 4.400 millones de años.
### Conceptos clave y resumen
Los núcleos que tienen cocientes n:p inestables sufren un decaimiento radiactivo espontáneo. Los tipos más comunes de radiactividad son el decaimiento α, el decaimiento β, la emisión γ, la emisión de positrones y la captura de electrones. En las reacciones nucleares también intervienen los rayos γ. Además, algunos núcleos decaen por captura de electrones. Cada uno de estos modos de decaimiento conduce a la formación de un nuevo núcleo con un cociente n:p más estable. Algunas sustancias sufren series de decaimiento radiactivo, ya que pasan por varios decaimientos antes de terminar en un isótopo estable. Todos los procesos de decaimiento nuclear siguen una cinética de primer orden, y cada radioisótopo tiene su propia semivida, el tiempo necesario para que la mitad de sus átomos decaiga. Debido a las grandes diferencias de estabilidad entre los núclidos, existe una gama muy amplia de semividas en las sustancias radiactivas. Muchas de estas sustancias tienen aplicaciones útiles en el diagnóstico y el tratamiento médico, en la determinación de la edad de los objetos arqueológicos y geológicos, y más.
### Ecuaciones clave
### Ejercicios de química del final del capítulo
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# Química nuclear
## Transmutación y energía nuclear
Tras el descubrimiento de la radiactividad, se creó el campo de la química nuclear, que se desarrolló rápidamente a principios del siglo XX. Una serie de nuevos descubrimientos en las décadas de 1930 y 1940, junto con la Segunda Guerra Mundial, se combinaron para dar paso a la Era Nuclear a mediados del siglo XX. Los científicos aprendieron a crear nuevas sustancias y se descubrió que ciertos isótopos de determinados elementos poseían la capacidad de producir cantidades de energía sin precedentes, con el potencial de causar enormes daños durante la guerra, así como de producir enormes cantidades de energía para las necesidades de la sociedad durante la paz.
### Síntesis de nucleidos
La transmutación nuclear es la conversión de un nucleido en otro. Se produce por el decaimiento radiactivo de un núcleo o por la reacción de un núcleo con otra partícula. El primer núcleo artificial se produjo en el laboratorio de Ernest Rutherford en 1919 mediante una reacción de transmutación: el bombardeo de un tipo de núcleo con otros núcleos o con neutrones. Rutherford bombardeó átomos de nitrógeno con partículas α de alta velocidad procedentes de un isótopo radiactivo natural de radio y observó los protones resultantes de la reacción:
Los núcleos y que se producen son estables, por lo que no ocurren más cambios (nucleares).
Para alcanzar las energías cinéticas necesarias para generar reacciones de transmutación, se utilizan dispositivos llamados aceleradores de partículas. Estos dispositivos utilizan campos magnéticos y eléctricos para aumentar la velocidad de las partículas nucleares. En todos los aceleradores, las partículas se mueven en el vacío para no colisionar con las moléculas de gas. Cuando se necesitan neutrones para las reacciones de transmutación, se obtienen de reacciones de decaimiento radiactivo o de diversas reacciones nucleares que se producen en los reactores nucleares. El siguiente artículo de "La química en la vida cotidiana” trata de un famoso acelerador de partículas que fue noticia en todo el mundo.
Antes de 1940, el elemento más pesado conocido era el uranio, cuyo número atómico es 92. Ahora, se han sintetizado y aislado muchos elementos artificiales, entre ellos varios a tan gran escala que han tenido un profundo efecto en la sociedad. Uno de ellos [el elemento 93, el neptunio (Np)] fue fabricado por primera vez en 1940 por McMillan y Abelson al bombardear uranio-238 con neutrones. La reacción crea uranio-239 inestable, con una semivida de 23,5 minutos, que luego decae en neptunio-239. El neptunio-239 también es radiactivo, con una semivida de 2,36 días, y decae en plutonio-239. Las reacciones nucleares son:
En la actualidad, el plutonio se forma principalmente en los reactores nucleares como subproducto durante la fisión del U-235. Durante este proceso de fisión se liberan neutrones adicionales (vea la siguiente sección), algunos de los cuales se combinan con núcleos de U-238 para formar uranio-239; este se somete a decaimiento β para formar neptunio-239, que a su vez se somete a decaimiento β para formar plutonio-239, como se ilustra en las tres ecuaciones anteriores. Estos procesos se resumen en la ecuación:
Los isótopos más pesados del plutonio (Pu-240, Pu-241 y Pu-242) también se producen cuando los núcleos más ligeros de plutonio capturan neutrones. Una parte de este plutonio altamente radiactivo se utiliza para fabricar armas militares; el resto representa un grave problema de almacenamiento porque tiene semividas de miles a cientos de miles de años.
Aunque no se han preparado en la misma cantidad que el plutonio, se han producido muchos otros núcleos sintéticos. La medicina nuclear se ha desarrollado a partir de la capacidad de convertir átomos de un tipo en otros tipos de átomos. Actualmente se utilizan isótopos radiactivos de varias docenas de elementos para aplicaciones médicas. La radiación producida por su decaimiento se utiliza para obtener imágenes o tratar diversos órganos o partes del cuerpo, entre otros usos.
Los elementos más allá del elemento 92 (uranio) se denominan elementos transuránicos. Al momento de redactar este artículo se han producido 22 elementos transuránicos, reconocidos oficialmente por la IUPAC). Otros elementos tienen reivindicaciones de formación que están a la espera de aprobación. Algunos de estos elementos se muestran en la .
### Fisión nuclear
Muchos elementos más pesados con energías de enlace más pequeñas por nucleón pueden descomponerse en elementos más estables que tienen números de masa intermedios y energías de enlace más grandes por nucleón, es decir, números de masa y energías de enlace por nucleón que están más cerca del "pico" del gráfico de energía de enlace cerca de 56 (ver la ). A veces también se producen neutrones. Esta descomposición se denomina fisión, es decir, la ruptura de un núcleo grande en trozos más pequeños. La ruptura es más bien aleatoria con la formación de un gran número de productos diferentes. La fisión no se produce de forma natural, sino que se induce mediante el bombardeo con neutrones. La primera fisión nuclear de la que se tiene constancia se produjo en 1939, cuando tres científicos alemanes, Lise Meitner, Otto Hahn y Fritz Strassman, bombardearon átomos de uranio-235 con neutrones de movimiento lento que dividieron los núcleos de U-238 en fragmentos más pequeños, formados por varios neutrones y elementos cercanos a la mitad de la tabla periódica. Desde entonces, se ha observado la fisión en muchos otros isótopos, incluso la mayoría de los isótopos de actínidos que tienen un número impar de neutrones. En la se observa una reacción de fisión nuclear.
Entre los productos de la reacción de fisión de Meitner, Hahn y Strassman se encontraban el bario, el criptón, el lantano y el cerio, todos ellos con núcleos más estables que el uranio-235. Desde entonces, se han observado cientos de isótopos diferentes entre los productos de las sustancias fisionables. Algunas de las muchas reacciones que se producen para el U-235, y un gráfico que muestra la distribución de sus productos de fisión y sus rendimientos, se muestran en la . Se han observado reacciones de fisión similares con otros isótopos del uranio, así como con una variedad de otros isótopos como los del plutonio.
La fisión de elementos pesados produce una enorme cantidad de energía. Por ejemplo, cuando un mol de U-235 sufre una fisión, los productos pesan unos 0,2 gramos menos que los reactivos; esta masa "perdida" se convierte en una cantidad muy grande de energía, unos 1,8 1010 kJ por mol de U-235. Las reacciones de fisión nuclear producen ingentes cantidades de energía en comparación con las reacciones químicas. La fisión de 1 kilogramo de uranio-235, por ejemplo, produce aproximadamente 2,5 millones de veces más energía que la producida por la combustión de 1 kilogramo de carbón.
Como se ha descrito anteriormente, al someterse a la fisión el U-235 produce dos núcleos "de tamaño medio" y dos o tres neutrones. Estos neutrones pueden provocar la fisión de otros átomos de uranio-235, que a su vez proporcionan más neutrones que pueden provocar la fisión de aún más núcleos, y así sucesivamente. Si esto ocurre, tenemos una reacción en cadena (vea la ). Por otro lado, si demasiados neutrones escapan del material en bruto sin interactuar con un núcleo, no se producirá ninguna reacción en cadena.
Se dice que el material que puede mantener una reacción nuclear en cadena de fisión es fisionable o fisible. (Técnicamente, el material fisible puede sufrir fisión con neutrones de cualquier energía, mientras que el material fisible requiere neutrones de alta energía). La fisión nuclear se convierte en autosostenible cuando el número de neutrones producidos por la fisión es igual o superior al número de neutrones absorbidos por los núcleos en división más el número que escapa a los alrededores. La cantidad de un material fisible que soportará una reacción en cadena autosostenida es una masa crítica. La cantidad de material fisible que no puede mantener una reacción en cadena es una masa subcrítica. La cantidad de material en la que hay una tasa de fisión creciente se conoce como masa supercrítica. La masa crítica depende del tipo de material: su pureza, la temperatura, la forma de la muestra y cómo se controlan las reacciones neutrónicas ().
Una bomba atómica () contiene varios kilos de material fisible, o una fuente de neutrones, y un dispositivo explosivo para comprimirla rápidamente en un pequeño volumen. Cuando el material fisible está en trozos pequeños, la proporción de neutrones que escapan a través de la superficie relativamente grande es grande, y no se produce ninguna reacción en cadena. Cuando los pequeños trozos de material fisible se juntan rápidamente para formar un cuerpo con una masa mayor que la masa crítica, el número relativo de neutrones que escapan disminuye, y se produce una reacción en cadena y una explosión.
### Reactores de fisión
Las reacciones en cadena de materiales fisibles pueden controlarse y sostenerse sin ninguna explosión en un reactor nuclear (). Todo reactor nuclear que produzca energía mediante la fisión de uranio o plutonio por bombardeo de neutrones deberá tener al menos cinco componentes: combustible nuclear formado por material fisionable, un moderador nuclear, refrigerante del reactor, varillas de control y un sistema de blindaje y contención. Más adelante hablaremos de estos componentes con más detalle. El reactor funciona al separar el material nuclear fisible de forma que no se pueda formar una masa crítica, controlando tanto el flujo como la absorción de neutrones para permitir el cierre de las reacciones de fisión. En un reactor nuclear utilizado para la producción de electricidad, la energía liberada por las reacciones de fisión queda atrapada como energía térmica y se utiliza para hervir agua y producir vapor. El vapor se utiliza para hacer girar una turbina, que acciona un generador para la producción de electricidad.
### Combustibles nucleares
El combustible nuclear consiste en un isótopo fisible, como el uranio-235, que deberá estar presente en cantidad suficiente para proporcionar una reacción en cadena autosostenible. En los Estados Unidos, los minerales de uranio contienen entre un 0,05 y un 0,3 % del óxido de uranio U3O8; el uranio del mineral es aproximadamente un 99,3 % de U-238 no fisible y solo un 0,7 % de U-235 fisible. Los reactores nucleares requieren un combustible con una concentración de U-235 superior a la que se encuentra en la naturaleza; normalmente se enriquece para que tenga alrededor del 5 % de la masa de uranio como U-235. Con esta concentración, no es posible alcanzar la masa supercrítica necesaria para una explosión nuclear. El uranio puede enriquecerse por difusión gaseosa (el único método que se utiliza actualmente en los EE. UU.), mediante una centrifugadora de gas o por separación láser.
En la planta de enriquecimiento por difusión gaseosa donde se prepara el combustible U-235, el gas UF6 (hexafluoruro de uranio) a baja presión se mueve a través de barreras que tienen agujeros apenas lo suficientemente grandes para que el UF6 pase. Las moléculas de 235UF6, un poco más ligeras, se difunden a través de la barrera con mayor rapidez que las moléculas de 238UF6, más pesadas. Este proceso se repite a través de cientos de barreras, aumentando gradualmente la concentración de 235UF6 hasta el nivel que necesita el reactor nuclear. La base de este proceso, la ley de Graham, se describe en el capítulo sobre los gases. El gas UF6 enriquecido se recoge, se enfría hasta que se solidifica y se lleva a una instalación de fabricación donde se convierte en elementos combustibles. Cada elemento combustible está formado por varillas de combustible que contienen muchos gránulos de combustible de uranio enriquecido (normalmente UO2) del tamaño de un dedal. Los reactores nucleares modernos pueden contener hasta 10 millones de gránulos de combustible. La cantidad de energía de cada uno de estos gránulos equivale a la de casi una tonelada de carbón o 150 galones de petróleo.
### Moderadores nucleares
Los neutrones que producen las reacciones nucleares se mueven demasiado rápido para provocar la fisión (vea la ). Primero deberán frenarse para que los absorba el combustible y así producir reacciones nucleares adicionales. El moderador nuclear es una sustancia que ralentiza los neutrones hasta una velocidad lo suficientemente baja como para provocar la fisión. Los primeros reactores utilizaban grafito de gran pureza como moderador. Los reactores modernos de los EE. UU. utilizan exclusivamente agua pesada o agua ligera (H2O ordinario), mientras que algunos reactores de otros países utilizan otros materiales, como dióxido de carbono, berilio o grafito.
### Refrigerantes del reactor
El refrigerante del reactor nuclear se utiliza para transportar el calor producido por la reacción de fisión a una caldera y una turbina externas, donde se transforma en electricidad. A menudo se utilizan dos bucles de refrigerante superpuestos; esto contrarresta la transferencia de radiactividad del reactor al bucle de refrigerante primario. Todas las centrales nucleares de los EE. UU. utilizan agua como refrigerante. Otros refrigerantes son el sodio fundido, el plomo, una mezcla de plomo y bismuto o las sales fundidas.
### Varillas de control
Los reactores nucleares utilizan varillas de control () para controlar la tasa de fisión del combustible nuclear al ajustar el número de neutrones lentos presentes para mantener la tasa de la reacción en cadena en un nivel seguro. Las varillas de control están hechas de boro, cadmio, hafnio u otros elementos capaces de absorber neutrones. El boro-10, por ejemplo, absorbe neutrones mediante una reacción que produce litio-7 y partículas alfa:
Cuando los conjuntos de varillas de control se insertan en el elemento de combustible del núcleo del reactor, absorben una mayor fracción de los neutrones lentos, con lo que se ralentiza el ritmo de la reacción de fisión y se reduce la potencia producida. Por el contrario, si se retiran las varillas de control, se absorben menos neutrones y aumentan la tasa de fisión y la producción de energía. En caso de emergencia, la reacción en cadena se corta al introducir completamente todas las varillas de control en el núcleo nuclear entre las varillas de combustible.
### Sistema de blindaje y contención
Un reactor nuclear en funcionamiento genera neutrones y otras radiaciones. Incluso cuando está apagado, los productos de decaimiento son radiactivos. Además, un reactor en funcionamiento está térmicamente muy caliente, y las altas presiones resultan de la circulación de agua u otro refrigerante a través de este. Por lo tanto, el reactor deberá soportar altas temperaturas y presiones, así como proteger a los operarios de la radiación. Los reactores están equipados con un sistema de contención (o escudo) que consta de tres partes:
1. El recipiente del reactor, una carcasa de acero de entre 3 y 20 centímetros de espesor que, junto con el moderador, absorbe gran parte de la radiación producida por el reactor.
2. Un escudo principal de 1 a 3 metros de hormigón de alta densidad.
3. Un escudo para el personal, hecho de un material más ligero, que protege a los operarios de los rayos γ, así como de los rayos X.
Además, los reactores suelen estar cubiertos por una cúpula de acero u hormigón, diseñada para contener cualquier material radiactivo que pueda liberarse por un accidente del reactor.
Las centrales nucleares están diseñadas de tal manera que no puedan formar ninguna masa supercrítica de material fisionable y, por ende, no puedan crear ninguna explosión nuclear. No obstante, como lo ha demostrado la historia, las fallas en los sistemas y las salvaguardias pueden provocar accidentes catastróficos, como explosiones químicas y fusiones nucleares (daños en el núcleo del reactor por recalentamiento). El siguiente artículo de “La química en la vida cotidiana” explora tres incidentes infames de fusión.
La energía que genera un reactor alimentado con uranio enriquecido es el resultado de la fisión del uranio, así como de la fisión del plutonio que se produce durante el funcionamiento del reactor. Como se ha comentado anteriormente, el plutonio se forma a partir de la combinación de neutrones y el uranio del combustible. En cualquier reactor nuclear, apenas el 0,1 % de la masa del combustible se convierte en energía. El otro 99,9 % permanece en las varillas de combustible como productos de fisión y combustible no utilizado. Todos los productos de fisión absorben neutrones y, tras un periodo que se prolonga desde varios meses hasta varios años, según el reactor, los productos de fisión deberán eliminarse al cambiar las varillas de combustible. De lo contrario, la concentración de estos productos de fisión aumentaría y absorbería más neutrones hasta que el reactor no pudiera seguir funcionando.
Las varillas de combustible gastadas contienen una variedad de productos, que consisten en núcleos inestables cuyo número atómico oscila entre 25 y 60, algunos elementos transuránicos, incluidos el plutonio y el americio, e isótopos de uranio sin reaccionar. Los núcleos inestables y los isótopos transuránicos confieren al combustible gastado un nivel de radiactividad peligrosamente alto. Los isótopos de larga vida necesitan miles de años para decaer hasta un nivel seguro. El destino final del reactor nuclear como fuente importante de energía en los Estados Unidos depende probablemente de que se pueda desarrollar una técnica política y científicamente satisfactoria para procesar y almacenar los elementos de las varillas de combustible gastado.
### Fusión nuclear y reactores de fusión
El proceso de conversión de núcleos muy ligeros en núcleos más pesados también va acompañado de la conversión de masa en grandes cantidades de energía. Dicho proceso recibe el nombre de fusión. La principal fuente de energía del sol es una reacción de fusión neta en la que cuatro núcleos de hidrógeno se fusionan y producen un núcleo de helio y dos positrones. Se trata de una reacción neta de una serie de acontecimientos más complicados:
Un núcleo de helio tiene una masa que es un 0,7 % menor que la de cuatro núcleos de hidrógeno; esta masa perdida se convierte en energía durante la fusión. Esta reacción produce alrededor de 3,6 1011 kJ de energía por mol de producido. Esto es algo mayor que la energía producida por la fisión nuclear de un mol de U-235 (1,8 1010 kJ), y más de 3 millones de veces mayor que la energía producida por la combustión (química) de un mol de octano (5471 kJ).
Se ha determinado que los núcleos de los isótopos pesados del hidrógeno, un deuterón, y un tritón, se fusionan a temperaturas extremadamente altas (fusión termonuclear). Forman un núcleo de helio y un neutrón:
Este cambio procede con una pérdida de masa de 0,0188 u, que corresponde a la liberación de 1,69 109 kilojulios por mol de formado. La altísima temperatura es necesaria para que los núcleos tengan la suficiente energía cinética para superar las fortísimas fuerzas de repulsión resultantes de las cargas positivas de sus núcleos y puedan colisionar.
Las reacciones de fusión útiles requieren temperaturas muy elevadas para su iniciación: unos 15.000.000 K o más. A estas temperaturas, todas las moléculas se disocian en átomos y estos se ionizan, formando el plasma. Estas condiciones se dan en un número extremadamente grande de lugares en todo el universo: las estrellas se alimentan de la fusión. Los seres humanos ya han descubierto cómo crear temperaturas lo suficientemente altas como para lograr la fusión a gran escala en las armas termonucleares. Un arma termonuclear, como una bomba de hidrógeno, contiene una bomba de fisión nuclear que, al explotar, desprende suficiente energía para producir las altísimas temperaturas necesarias para que se produzca la fusión.
Otra forma mucho más beneficiosa de crear reacciones de fusión es en un reactor de fusión, un reactor nuclear en el que se controlan las reacciones de fusión de núcleos ligeros. Dado que ningún material sólido es estable a tan altas temperaturas, los dispositivos mecánicos no pueden contener el plasma en el que se producen las reacciones de fusión. Dos técnicas para contener el plasma a la densidad y temperatura necesarias para una reacción de fusión son actualmente objeto de intensos esfuerzos de investigación: la contención mediante un campo magnético y el uso de rayos láser enfocados (). Varios proyectos de gran envergadura trabajan para alcanzar uno de los mayores objetivos de la ciencia: conseguir que el combustible de hidrógeno se encienda y produzca más energía que la suministrada para alcanzar las altísimas temperaturas y presiones que se requieren para la fusión. Al momento de redactar este artículo, no hay reactores de fusión autosostenibles en funcionamiento en el mundo, aunque se han llevado a cabo reacciones de fusión controladas a pequeña escala durante periodos muy breves.
### Conceptos clave y resumen
Es posible producir nuevos átomos con el bombardeo de otros átomos con núcleos o partículas de alta velocidad. Los productos de estas reacciones de transmutación pueden ser estables o radiactivos. Se han producido de este modo varios elementos artificiales, como el tecnecio, la astatina y los elementos transuránicos.
La energía nuclear, así como las detonaciones de armas nucleares, pueden generarse mediante la fisión (reacciones en las que un núcleo pesado se divide en dos o más núcleos más ligeros y varios neutrones). Dado que los neutrones inducen reacciones de fisión adicionales al combinarse con otros núcleos pesados, se produciría una reacción en cadena. La energía útil se obtiene si el proceso de fisión se lleva a cabo en un reactor nuclear. La conversión de núcleos ligeros en núcleos más pesados (fusión) también genera energía. En la actualidad, esta energía no se ha contenido adecuadamente y es demasiado cara como para ser viable para la producción de energía comercial.
### Ejercicios de química del final del capítulo
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# Química nuclear
## Usos de los radioisótopos
Los isótopos radiactivos tienen las mismas propiedades químicas que los isótopos estables del mismo elemento, pero emiten radiaciones que pueden detectarse. Si sustituimos uno o más átomos por radioisótopos en un compuesto, podemos rastrearlos por sus emisiones radiactivas. Este tipo de compuesto se denomina trazador radiactivo (o etiqueta radiactiva). Los radioisótopos se utilizan para seguir las rutas de las reacciones bioquímicas o para determinar la distribución de una sustancia en un organismo. Asimismo, los trazadores radiactivos se utilizan en muchas aplicaciones médicas, tanto en el diagnóstico como en el tratamiento. Se emplean para medir el desgaste de los motores, analizar la formación geológica alrededor de los pozos de petróleo y mucho más.
Los radioinmunoensayos (RIA), por ejemplo, se basan en radioisótopos para detectar la presencia o la concentración de determinados antígenos. Esta técnica, desarrollada por Rosalyn Sussman Yalow y Solomon Berson en la década de 1950, es conocida por su extrema sensibilidad, lo que significa que puede detectar y medir cantidades muy pequeñas de una sustancia. Antes de su descubrimiento, la mayoría de las detecciones similares se basaban en cantidades suficientemente grandes para producir resultados visibles. Los RIA revolucionaron y ampliaron campos enteros de estudio, sobre todo la endocrinología, y se utilizan habitualmente en la detección de narcóticos, la detección en bancos de sangre, la detección precoz del cáncer, la medición de hormonas y el diagnóstico de alergias. Gracias a su importante contribución a la medicina, Yalow recibió el Premio Nobel, lo que la convirtió en la segunda mujer galardonada con este premio en el área de medicina.
Los radioisótopos han revolucionado la práctica médica (vea el Apéndice M), donde se utilizan ampliamente. En los Estados Unidos se realizan anualmente más de 10 millones de procedimientos y más de 100 millones de pruebas de medicina nuclear. Cuatro ejemplos típicos de trazadores radiactivos utilizados en medicina son el tecnecio-99 , el talio-201 , el yodo-131 y el sodio-24 . Los tejidos dañados del corazón, del hígado y de los pulmones absorben preferentemente ciertos compuestos de tecnecio-99. Tras su inyección, la localización del compuesto de tecnecio, y por tanto del tejido dañado, se determina con la detección de los rayos γ emitidos por el isótopo Tc-99. El talio-201 () se concentra en el tejido cardíaco sano, por lo que los dos isótopos, Tc-99 y Tl-201, se utilizan juntos para estudiar el tejido cardíaco. El yodo-131 se concentra en la glándula tiroides, el hígado y algunas partes del cerebro. Por consiguiente, se utiliza para controlar el bocio y tratar las afecciones tiroideas, como la enfermedad de Grave, así como los tumores hepáticos y cerebrales. Las soluciones salinas que contienen compuestos de sodio-24 se inyectan en el torrente sanguíneo para localizar las obstrucciones del flujo sanguíneo.
Los radioisótopos utilizados en medicina tienen semividas cortas; por ejemplo, el omnipresente Tc-99m tiene una semivida de 6,01 horas. Esto hace que el Tc-99m sea esencialmente imposible de almacenar y prohibitivamente caro de transportar, por lo que se fabrica in situ. Los hospitales y otras instalaciones médicas utilizan el Mo-99 (que se extrae principalmente de los productos de fisión del U-235) para generar Tc-99. El Mo-99 sufre un decaimiento β con una semivida de 66 horas, y el Tc-99 se extrae entonces químicamente (). El nucleido padre Mo-99 forma parte de un ion de molibdato, cuando decae, forma el ion de pertecnetato, Estos dos iones solubles en agua se separan por cromatografía en columna, en la que el ion de molibdato de mayor carga se adsorbe en la alúmina de la columna, y el ion de pertecnetato de menor carga pasa por la columna en la solución. Unos cuantos microgramos de Mo-99 producen suficiente Tc-99 para realizar hasta 10.000 pruebas.
Los radioisótopos también se utilizan como tratamiento, normalmente en dosis más altas que como trazador. La radioterapia es el uso de radiación potente para dañar el ADN de las células cancerosas, lo que las mata o impide que se dividan (). Un paciente con cáncer puede recibir radioterapia de haz externo administrada por una máquina fuera del cuerpo, o radioterapia interna (braquiterapia) a partir de una sustancia radiactiva que se introduce en el organismo. Tenga en cuenta que la quimioterapia es semejante a la radioterapia interna en el sentido de que el tratamiento oncológico se inyecta en el organismo, pero difiere en que la quimioterapia utiliza sustancias químicas en lugar de radiactivas para eliminar las células cancerosas.
El cobalto-60 es un radioisótopo sintético que se produce por la activación neutrónica del Co-59, que luego sufre un decaimiento β para formar Ni-60, junto con la emisión de radiación γ. El proceso general es:
El esquema general de decaimiento de esto se muestra gráficamente en la .
Los radioisótopos se utilizan de diversas maneras para estudiar los mecanismos de las reacciones químicas en plantas y animales. Entre ellos se encuentran el etiquetado de los fertilizantes en los estudios sobre la absorción de nutrientes por parte de las plantas y el crecimiento de los cultivos, las investigaciones sobre los procesos digestivos y de producción de leche en las vacas, y los estudios sobre el crecimiento y el metabolismo de los animales y de las plantas.
Por ejemplo, el radioisótopo C-14 se utilizó para dilucidar los detalles de la fotosíntesis. La reacción general es:
pero el proceso es mucho más complejo, ya que pasa por una serie de pasos en los que se producen diversos compuestos orgánicos. En los estudios sobre la vía de esta reacción, las plantas fueron expuestas a CO2 con alta concentración de . A intervalos regulares se analizaron las plantas para determinar qué compuestos orgánicos contenían carbono-14 y qué cantidad de cada compuesto estaba presente. A partir de la secuencia temporal en la que aparecieron los compuestos y la cantidad de cada uno de estos en determinados intervalos, los científicos aprendieron más sobre la vía de la reacción.
Las aplicaciones comerciales de los materiales radiactivos son igualmente diversas (). Entre estas se encuentra la determinación del grosor de las películas y de las láminas metálicas delgadas aprovechando el poder de penetración de variados tipos de radiación. Los defectos en los metales que se emplean con fines estructurales se detectan con potentes rayos gama, provenientes del cobalto 60, parecido a la forma en que se utilizan los rayos X para examinar el cuerpo humano. En una forma de control de plagas, las moscas se controlan esterilizando a los machos con radiación γ para que las hembras que se reproduzcan con ellos no produzcan descendencia. Muchos alimentos se conservan mediante radiaciones que matan los microorganismos causantes de la descomposición de los alimentos.
El americio-241, un emisor α con semivida de 458 años, se utiliza en cantidades ínfimas en los detectores de humo de tipo ionizante (). Las emisiones α de Am-241 ionizan el aire entre dos placas de electrodos en la cámara de ionización. Una batería suministra un potencial que provoca el movimiento de los iones, para generar una pequeña corriente eléctrica. Cuando el humo entra en la cámara, se impide el movimiento de los iones, lo que reduce la conductividad del aire. Esto provoca una caída ostensible en la corriente y acciona una alarma.
### Conceptos clave y resumen
Los compuestos conocidos como trazadores radiactivos se emplean para seguir reacciones, rastrear la distribución de una sustancia, diagnosticar y tratar enfermedades y mucho más. Otras sustancias radiactivas sirven para controlar las plagas, visualizar las estructuras, proporcionar avisos de incendio y en infinidad de aplicaciones. En los Estados Unidos se realizan cada año cientos de millones de pruebas y procedimientos de medicina nuclear con una gran variedad de radioisótopos de semividas relativamente cortas. La mayoría de estos radioisótopos tienen semividas relativamente cortas; algunas son lo suficientemente cortas como para que el radioisótopo tenga que fabricarse in situ en las instalaciones médicas. La radioterapia consiste en radiación de alta energía para eliminar las células cancerosas al dañar su ADN. La radiación para este tratamiento se administra de forma externa o interna.
### Ejercicios de química del final del capítulo
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# Química nuclear
## Efectos biológicos de la radiación
El aumento del uso de radioisótopos ha dado lugar a una mayor preocupación por los efectos de estos materiales en los organismos biológicos (como en el caso del cuerpo humano). Todos los nucleidos radiactivos emiten partículas de alta energía u ondas electromagnéticas. Cuando esta radiación entra en contacto con las células vivas, puede causar calentamiento, romper los enlaces químicos o ionizar las moléculas. Los daños biológicos más graves se producen cuando estas emisiones radiactivas fragmentan o ionizan las moléculas. Por ejemplo, las partículas alfa y beta, emitidas en las reacciones de decaimiento nuclear, poseen energías mucho más altas que las energías de los enlaces químicos ordinarios. Cuando estas partículas golpean y penetran en la materia, producen iones y fragmentos moleculares que son extremadamente reactivos. El daño que esto provoca en las biomoléculas de los organismos vivos puede causar graves disfunciones en los procesos celulares normales, lo que pone a prueba los mecanismos de reparación del organismo y puede causar enfermedades y hasta la muerte ().
### Radiación ionizante y no ionizante
Existe una gran diferencia en la magnitud de los efectos biológicos de la radiación no ionizante (por ejemplo, la luz y las microondas) y de la radiación ionizante, emisiones lo suficientemente energéticas como para desprender electrones de las moléculas (por ejemplo, las partículas α y β, los rayos γ, los rayos X y la radiación ultravioleta de alta energía) ().
La energía que absorbe la radiación no ionizante acelera el movimiento de los átomos y las moléculas, lo que equivale a calentar la muestra. Aunque los organismos biológicos son sensibles al calor (como podemos saber por haber tocado una estufa caliente o haber pasado un día de playa al sol), es necesaria una gran cantidad de radiación no ionizante antes de que se alcancen niveles peligrosos. Sin embargo, la radiación ionizante puede causar daños mucho más graves al romper los enlaces o eliminar los electrones de las moléculas biológicas, ya que altera su estructura y función. El daño también puede producirse de forma indirecta, al ionizar primero el H2O (la molécula más abundante en los organismos vivos), que forma un ion de H2O+ que reacciona con el agua, para formar un ion hidronio y un radical hidroxilo.
Debido a que el radical hidroxilo tiene un electrón desapareado, es muy reactivo. (Este es el caso en cualquier sustancia con electrones desapareados, conocida como radical libre). Este radical hidroxilo puede reaccionar con todo tipo de moléculas biológicas (ADN, proteínas, enzimas, etc.), y causar daños a las moléculas y alterar los procesos fisiológicos. En la se muestran ejemplos de daños directos e indirectos.
### Efectos biológicos de la exposición a la radiación
La radiación puede dañar todo el cuerpo (daño somático) o los óvulos y el esperma (daño genético). Sus efectos son más pronunciados en las células que se reproducen rápidamente, como el revestimiento del estómago, los folículos pilosos, la médula ósea y los embriones. Por ello, las pacientes sometidas a radioterapia suelen sentir náuseas o malestar estomacal, perder el cabello, tener dolores en los huesos, etc., y por ello hay que tener especial cuidado al someterse a la radioterapia durante el embarazo.
Los diferentes tipos de radiación tienen diferentes capacidades para atravesar el material (). Una barrera muy fina, como una o dos hojas de papel, o la capa superior de las células de la piel, suele detener las partículas alfa. Por este motivo, las fuentes de partículas alfa no son peligrosas si se encuentran fuera del cuerpo, pero son bastante peligrosas si se ingieren o inhalan (vea el artículo de Química en la vida cotidiana sobre la exposición al radón). Las partículas beta atraviesan una mano, o una fina capa de material como el papel o la madera, pero las detiene una fina capa de metal. La radiación gama es muy penetrante y puede atravesar una capa gruesa de la mayoría de los materiales. Algunas radiaciones gama de alta energía son capaces de atravesar algunos pies de hormigón. Algunos elementos densos y de alto número atómico (como el plomo) pueden atenuar eficazmente la radiación gama con material más fino y se utilizan para el blindaje. La capacidad de los distintos tipos de emisiones para provocar la ionización varía mucho, y algunas partículas casi no tienen tendencia a producir ionización. Las partículas alfa tienen aproximadamente el doble de poder ionizante que los neutrones rápidos, unas 10 veces más que las partículas β y unas 20 veces más que los rayos γ y los rayos X.
### Medición de la exposición a la radiación
Para detectar y medir las radiaciones se utilizan diferentes dispositivos, como los contadores Geiger, los contadores de centelleo (centelladores) y los dosímetros de radiación (). Probablemente el instrumento de radiación más conocido, el contador Geiger (también llamado contador Geiger-Müller) detecta y mide la radiación. La radiación provoca la ionización del gas en un tubo Geiger-Müller. La tasa de ionización es proporcional a la cantidad de radiación. El contador de centelleo contiene un centellador. Este consiste en un material que emite luz (luminiscencia) con el estímulo de la radiación ionizante y un sensor que convierte la luz en una señal eléctrica. Los dosímetros de radiación también miden la radiación ionizante y se utilizan para determinar la exposición personal a la radiación. Los tipos más utilizados son los dosímetros electrónicos, de lámina, termoluminiscentes y de fibra de cuarzo.
Se utiliza una variedad de unidades para medir diversos aspectos de la radiación (). La unidad del sistema internacional (SI) para la tasa de decaimiento radiactivo es el becquerel (Bq), donde 1 Bq = 1 decaimiento por segundo. El curio (Ci) y el milicurio (mCi) son unidades mucho mayores y se utilizan con frecuencia en medicina (1 curio = 1 Ci = 3,7 1010 decaimientos por segundo). La unidad del SI para medir la dosis de radiación es el gray (Gy), donde 1 Gy = 1 J de energía absorbida por kilogramo de tejido. En las aplicaciones médicas, se utiliza más a menudo la dosis de radiación absorbida (rad) (1 rad = 0,01 Gy; 1 rad da lugar a la absorción de 0,01 J/kg de tejido). La unidad del SI que mide el daño tisular causado por la radiación es el sievert (Sv). Esto tiene en cuenta tanto la energía como los efectos biológicos del tipo de radiación implicado en la dosis de radiación. El equivalente en roentgen para el hombre (rem) es la unidad de daño por radiación que más se utiliza en medicina (100 rem = 1 Sv). Observe que las unidades de daño tisular (rem o Sv) incluyen la energía de la dosis de radiación (rad o Gy) junto con un factor biológico denominado RBE (de eficacia biológica relativa [relative biological effectiveness]) que es una medida aproximada del daño relativo que causa la radiación. Estos están relacionados por:
donde RBE es aproximadamente 10 para la radiación α, 2(+) para protones y neutrones, y 1 para la radiación β y la radiación γ.
### Unidades de medida de la radiación
La reseña las unidades que se utilizan para medir la radiación.
### Efectos en el cuerpo humano de la exposición prolongada a la radiación
Los efectos de la radiación dependen del tipo, la energía y la ubicación de la fuente de radiación, así como de la duración de la exposición. Como se muestra en la , la persona media está expuesta a la radiación de fondo, incluidos los rayos cósmicos del sol y el radón del uranio en el suelo (vea el artículo de Química en la vida cotidiana sobre la exposición al radón), la radiación de la exposición médica, incluidos la exploración por TAC, las pruebas de radioisótopos, los rayos X, etc., y pequeñas cantidades de radiación de otras actividades humanas, como los vuelos de aviones (que son bombardeados por un mayor número de rayos cósmicos en la atmósfera superior), la radiactividad de los productos de consumo y una variedad de radionúclidos que entran en nuestro cuerpo cuando respiramos (por ejemplo, el carbono-14) o a través de la cadena alimentaria (por ejemplo, el potasio-40, el estroncio-90 y el yodo-131).
Una dosis repentina y de corta duración de una gran cantidad de radiación puede causar una amplia gama de efectos sobre la salud, desde cambios en la química de la sangre hasta la muerte. La exposición a corto plazo a decenas de rems de radiación probablemente causará síntomas o enfermedades muy notables; se estima que una dosis de unos 500 rems tiene una probabilidad del 50 % de causar la muerte de la víctima en los 30 días siguientes a la exposición. La exposición a las emisiones radiactivas tiene un efecto acumulativo en el organismo a lo largo de la vida de una persona, lo que constituye otra razón por la que es importante evitar cualquier exposición innecesaria a la radiación. Los efectos sobre la salud de la exposición a corto plazo a la radiación se muestran en la .
Es imposible evitar cierta exposición a las radiaciones ionizantes. Estamos expuestos constantemente a la radiación espontánea de diversas fuentes naturales, como la radiación cósmica, las rocas, los procedimientos médicos, los bienes de consumo e incluso nuestros propios átomos. Podemos minimizar nuestra exposición al bloquear o protegernos de la radiación, alejarnos de la fuente y limitar el tiempo de exposición.
### Conceptos clave y resumen
Estamos expuestos constantemente a las radiaciones procedentes de diversas fuentes naturales y producidas por el hombre. Esta radiación puede afectar a los organismos vivos. La radiación ionizante es la más perjudicial, ya que puede ionizar las moléculas o romper los enlaces químicos, lo que daña la molécula y provoca mal funcionamiento en los procesos celulares. También puede crear radicales hidroxilos reactivos que dañan las moléculas biológicas y alteran los procesos fisiológicos. La radiación puede causar daños somáticos o genéticos, y es más perjudicial para las células que se reproducen rápidamente. Los tipos de radiación difieren en su capacidad de penetrar en el material y dañar el tejido: las partículas alfa son las menos penetrantes, aunque potencialmente más dañinas; los rayos gamma son los más penetrantes.
Para detectar y medir la radiación y controlar la exposición se utilizan diversos dispositivos, como contadores Geiger, centelladores y dosímetros. Utilizamos varias unidades para medir la radiación: becquerel o curios para la tasa de decaimiento radiactivo; gray o rads para la energía absorbida, y rems o sieverts para los efectos biológicos de la radiación. La exposición a la radiación causa una amplia gama de efectos sobre la salud, que van desde leves hasta graves, e incluso la muerte. Podemos minimizar los efectos de la radiación si nos protegemos con materiales densos como el plomo, nos alejamos de la fuente y limitamos el tiempo de exposición.
### Ecuaciones clave
### Ejercicios de química del final del capítulo
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# Química orgánica
## Introducción
Todos los seres vivos de la Tierra están formados principalmente por compuestos de carbono. La prevalencia de los compuestos de carbono en los seres vivos ha dado lugar al epíteto de vida "basada en el carbono". La verdad es que no conocemos otro tipo de vida. Los primeros químicos consideraban que las sustancias aisladas de los organismos (plantas y animales) eran un tipo distinto de materia que no podía sintetizarse artificialmente, por lo que estas sustancias se conocían como compuestos orgánicos. La creencia generalizada, denominada vitalismo, sostenía que los compuestos orgánicos estaban formados por una fuerza vital presente solo en los organismos vivos. El químico alemán Friedrich Wohler fue uno de los primeros en refutar este aspecto del vitalismo, cuando, en 1828, informó acerca de la síntesis de la urea, un componente de muchos fluidos corporales, a partir de materiales no vivos. Desde entonces, se ha reconocido que las moléculas orgánicas obedecen a las mismas leyes naturales que las sustancias inorgánicas, y la categoría de compuestos orgánicos ha evolucionado para incluir tanto los compuestos naturales como los sintéticos que contienen carbono. Algunos compuestos que contienen carbono no se clasifican como orgánicos, por ejemplo, los carbonatos y cianuros, y los óxidos simples, como el CO y el CO2. Aunque el gremio químico aún no ha dado ninguna definición única y precisa, la mayoría coincide en que un rasgo definitorio de las moléculas orgánicas es la presencia de carbono como elemento principal, unido a hidrógeno y a otros átomos de carbono.
En la actualidad, los compuestos orgánicos son elementos fundamentales de plásticos, jabones, perfumes, edulcorantes, tejidos, productos farmacéuticos y muchas otras sustancias que utilizamos a diario. El valor que tienen para nosotros los compuestos orgánicos hace que la química orgánica sea una disciplina importante dentro del campo general de la química. En este capítulo, analizamos por qué el elemento carbono da lugar a un gran número y variedad de compuestos, cómo se clasifican esos compuestos y el papel de los compuestos orgánicos en entornos biológicos e industriales representativos. |
# Química orgánica
## Hidrocarburos
La mayor base de datosSe trata de la base de datos Beilstein, disponible ahora a través del sitio web Reaxys (www.elsevier.com/online-tools/reaxys). de compuestos orgánicos enumera unos 10 millones de sustancias, que incluyen compuestos procedentes de organismos vivos y los sintetizados por los químicos. El número de compuestos orgánicos potenciales se calculaPeplow, Mark. “Organic Synthesis: The Robo-Chemist” ("Síntesis orgánica: el robot químico"), en 1060, una cifra estratosférica. La existencia de tantas moléculas orgánicas es consecuencia de la capacidad de los átomos de carbono de formar hasta cuatro enlaces fuertes con otros átomos de carbono, lo que da lugar a cadenas y anillos de muy diversos tamaños, formas y complejidades.
Los compuestos orgánicos más simples contienen únicamente los elementos carbono e hidrógeno, y se denominan hidrocarburos. Aunque están compuestos únicamente por dos tipos de átomos, existe una gran variedad de hidrocarburos porque pueden consistir en cadenas de distinta longitud, cadenas ramificadas y anillos de átomos de carbono o combinaciones de estas estructuras. Además, los hidrocarburos pueden diferir en los tipos de enlaces carbono-carbono que están presentes en sus moléculas. Muchos hidrocarburos se encuentran en las plantas, los animales y sus fósiles; otros se han preparado en el laboratorio. Todos los días utilizamos hidrocarburos, principalmente como combustible, tales como el gas natural, el acetileno, el propano, el butano y los principales componentes de la gasolina, el gasóleo y el gasóleo de calefacción. Los conocidos plásticos polietileno, polipropileno y poliestireno también son hidrocarburos. Podemos distinguir varios tipos de hidrocarburos por las diferencias en el enlace entre los átomos de carbono. Esto da lugar a diferencias en las geometrías y en la hibridación de los orbitales del carbono.
### Alcanos
Los alcanos, o hidrocarburos saturados, contienen solamente enlaces covalentes simples entre los átomos de carbono. Cada uno de los átomos de carbono de un alcano tiene orbitales híbridos sp3 y está enlazado a otros cuatro átomos, cada uno de los cuales es carbono o hidrógeno. Las estructuras de Lewis y los modelos de metano, etano y pentano se ilustran en la . Las cadenas de carbono suelen dibujarse como líneas rectas en las estructuras de Lewis, pero hay que recordar que estas no pretenden indicar la geometría de las moléculas. Observe que los átomos de carbono en los modelos estructurales (los modelos de barras y esferas, y de espacio lleno) de la molécula de pentano no se encuentran en línea recta. Debido a la hibridación sp3, los ángulos de enlace en las cadenas de carbono son cercanos a 109,5°, lo que da a dichas cadenas en un alcano una forma de zigzag.
Las estructuras de los alcanos y otras moléculas orgánicas también pueden representarse de forma menos detallada mediante fórmulas estructurales condensadas (o simplemente, fórmulas condensadas). En lugar del formato habitual de las fórmulas químicas, en el que cada símbolo de elemento aparece una sola vez, se escribe una fórmula condensada para sugerir el enlace en la molécula. Estas fórmulas tienen la apariencia de una estructura de Lewis de la que se han eliminado la mayoría o todos los símbolos de enlace. Las fórmulas estructurales condensadas del etano y el pentano se muestran en la parte inferior de la , y en los ejercicios que se encuentran al final de este capítulo se ofrecen varios otros ejemplos.
Un método que utilizan comúnmente los químicos orgánicos para simplificar los dibujos de moléculas más grandes es la estructura esquelética (también llamada estructura de ángulo de línea). En este tipo de estructura, los átomos de carbono no se simbolizan con una C, sino que se representan con cada extremo de una línea o curva en una línea. Los átomos de hidrógeno no se dibujan si están unidos a un carbono. Otros átomos, además del carbono y el hidrógeno, se representan con sus símbolos elementales. La muestra tres formas diferentes de dibujar la misma estructura.
Todos los alcanos están compuestos por átomos de carbono e hidrógeno, y tienen enlaces, estructuras y fórmulas similares; los alcanos no cíclicos tienen todos una fórmula de CnH2n+2. El número de átomos de carbono presentes en un alcano no tiene límite. Un mayor número de átomos en las moléculas dará lugar a atracciones intermoleculares más fuertes (fuerzas de dispersión) y a las correspondientes propiedades físicas distintas de las moléculas. Propiedades como el punto de fusión y el punto de ebullición () cambian de manera fluida y predecible al cambiar el número de átomos de carbono e hidrógeno en las moléculas.
Los hidrocarburos con la misma fórmula, incluso los alcanos, pueden tener estructuras diferentes. Por ejemplo, dos alcanos tienen la fórmula C4H10: Se llaman n-butano y 2-metilpropano (o isobutano), y tienen las siguientes estructuras de Lewis:
Los compuestos n-butano y 2-metilpropano son isómeros estructurales (también se utiliza el término isómeros constitucionales). Los isómeros constitucionales tienen la misma fórmula molecular, pero distintas disposiciones espaciales de los átomos en sus moléculas. La molécula n-butano contiene una cadena no ramificada, lo que significa que ningún átomo de carbono está unido a más de otros dos átomos de carbono. Utilizamos el término normal o el prefijo n, para referirnos a una cadena de átomos de carbono sin ramificaciones. El compuesto 2-metilpropano tiene una cadena ramificada (el átomo de carbono en el centro de la estructura de Lewis está unido a otros tres átomos de carbono).
Identificar los isómeros a partir de las estructuras de Lewis no es tan fácil como parece. Las estructuras de Lewis que lucen diferentes pueden representar en realidad los mismos isómeros. Por ejemplo, las tres estructuras en la representan la misma molécula, n-butano, y por ende, no son isómeros diferentes. Son idénticos porque cada uno contiene una cadena no ramificada de cuatro átomos de carbono.
### Los fundamentos de la nomenclatura orgánica: designación de los alcanos
La Unión Internacional de Química Pura y Aplicada (IUPAC) ha ideado un sistema de nomenclatura que comienza con los nombres de los alcanos y puede ajustarse a partir de ahí para tener en cuenta estructuras más complicadas. La nomenclatura de los alcanos se basa en dos reglas:
1. Para designar un alcano, primero hay que identificar la cadena más larga de átomos de carbono en su estructura. La cadena de dos carbonos se denomina etano; la de tres, propano, y la de cuatro, butano. Las cadenas más largas se denominan así: pentano (cadena de cinco carbonos), hexano (6), heptano (7), octano (8), nonano (9) y decano (10). Estos prefijos se observan en los nombres de los alcanos que se describen en la .
2. Añada prefijos al nombre de la cadena más larga para indicar las posiciones y los nombres de los sustituyentes. Los sustituyentes son ramas o grupos funcionales que sustituyen a los átomos de hidrógeno de una cadena. La posición de un sustituyente o rama se identifica por el número del átomo de carbono al que está unido en la cadena. Numeramos los átomos de carbono de la cadena contando desde el extremo de la cadena más cercano a los sustituyentes. Los sustituyentes múltiples se nombran individualmente y se colocan en orden alfabético al principio del nombre.
Cuando hay más de un sustituyente, ya sea en el mismo átomo de carbono o en diferentes átomos de carbono, los sustituyentes se enumeran por orden alfabético. Dado que la numeración de los átomos de carbono comienza en el extremo más cercano a un sustituyente, la cadena más larga de átomos de carbono se numera de forma que se obtenga el número más bajo para los sustituyentes. La terminación -o sustituye a -uro al final del nombre de un sustituyente electronegativo (en los compuestos iónicos, el ion con carga negativa termina con -uro como el cloruro; en los compuestos orgánicos, estos átomos se tratan como sustituyentes y se utiliza la terminación -o ). El número de sustituyentes del mismo tipo se indica con los prefijos di- (dos), tri- (tres), tetra- (cuatro), etc. (por ejemplo, difluoro- indica dos sustituyentes del fluoruro).
Llamamos grupo alquilo a un sustituyente que contiene un hidrógeno menos que el alcano correspondiente. El nombre del grupo alquilo se obtiene al suprimir el sufijo -ano del nombre del alcano y añadir -ilo:
Los enlaces abiertos en los grupos metilo y etilo indican que estos grupos alquilo están enlazados a otro átomo.
Algunos hidrocarburos forman más de un tipo de grupo alquilo cuando los átomos de hidrógeno que se eliminarían tienen diferentes "ambientes" en la molécula. Esta diversidad de posibles grupos alquilos se identifica de la siguiente manera: Los cuatro átomos de hidrógeno de una molécula de metano son equivalentes; todos tienen el mismo ambiente. Son equivalentes porque cada uno está enlazado a un átomo de carbono (el mismo átomo de carbono), que está enlazado a tres átomos de hidrógeno. (Sería más fácil ver la equivalencia en los modelos de barras y esferas en la . La eliminación de uno de los cuatro átomos de hidrógeno del metano forma un grupo metilo. Del mismo modo, los seis átomos de hidrógeno del etano son equivalentes () y la eliminación de cualquiera de estos átomos de hidrógeno produce un grupo etilo. Cada uno de los seis átomos de hidrógeno está enlazado a un átomo de carbono, que está enlazado a otros dos átomos de hidrógeno y a un átomo de carbono. Sin embargo, tanto en el propano como en el 2-metilpropano, hay átomos de hidrógeno en dos ambientes diferentes, que se distinguen por los átomos o grupos de átomos adyacentes:
Cada uno de los seis átomos de hidrógeno equivalentes del primer tipo en el propano y cada uno de los nueve átomos de hidrógeno equivalentes de ese tipo en el 2-metilpropano (todos mostrados en negro) están enlazados a un átomo de carbono que está enlazado a otro átomo de carbono. Los dos átomos de hidrógeno en púrpura del propano son de un segundo tipo. Se distinguen de los seis átomos de hidrógeno del primer tipo en que están enlazados a un átomo de carbono enlazado a otros dos átomos de carbono. El átomo de hidrógeno en verde del 2-metilpropano se distingue de los otros nueve átomos de hidrógeno de esa molécula y de los átomos de hidrógeno en púrpura del propano. El átomo de hidrógeno en verde del 2-metilpropano está enlazado a un átomo de carbono enlazado a otros tres átomos de carbono. A partir de cada una de estas moléculas se forman dos distintos grupos alquílicos, dependiendo del átomo de hidrógeno que se elimine. Los nombres y las estructuras de estos y otros grupos alquílicos se enumeran en la .
Observe que los grupos alquilos no existen como entidades estables independientes. Siempre forman parte de una molécula mayor. La ubicación de un grupo alquilo en una cadena de hidrocarburos se indica de la misma manera que cualquier otro sustituyente:
Los alcanos son moléculas relativamente estables, pero el calor o la luz activan reacciones que implican la ruptura de enlaces simples C-H o C-C. La combustión es una de estas reacciones:
Los alcanos arden en presencia de oxígeno, una reacción de oxidación-reducción altamente exotérmica que produce dióxido de carbono y agua. En consecuencia, los alcanos son excelentes combustibles. Por ejemplo, el metano, CH4, es el principal componente del gas natural. El butano, C4H10, utilizado en las estufas y encendedores de camping, es un alcano. La gasolina es una mezcla líquida de alcanos de cadena continua y ramificada, cada uno de los cuales contiene de cinco a nueve átomos de carbono, más diversos aditivos para mejorar su rendimiento como combustible. El queroseno, el gasóleo y el combustóleo son principalmente mezclas de alcanos con masas moleculares más altas. La principal fuente de estos combustibles alcalinos líquidos es el petróleo crudo, una mezcla compleja que se separa por destilación fraccionada. La destilación fraccionada aprovecha las diferencias en los puntos de ebullición de los componentes de la mezcla (vea la ). Recordará que el punto de ebullición es una función de las interacciones intermoleculares, que se trató en el capítulo sobre soluciones y coloides.
En una reacción de sustitución, otra reacción típica de los alcanos, uno o varios de los átomos de hidrógeno del alcano se sustituyen por un átomo o grupo de átomos diferente. En estas reacciones no se rompen los enlaces carbono-carbono y la hibridación de los átomos de carbono no cambia. Por ejemplo, la reacción entre el etano y el cloro molecular representada aquí es una reacción de sustitución:
La porción C-Cl de la molécula de cloroetano es un ejemplo de grupo funcional, la parte o fracción de una molécula que imparte una reactividad química específica. Los tipos de grupos funcionales presentes en una molécula orgánica son los principales determinantes de sus propiedades químicas y se utilizan como medio de clasificación de los compuestos orgánicos, tal y como se detalla en las restantes secciones de este capítulo.
### Alquenos
Los compuestos orgánicos que contienen uno o más enlaces dobles o triples entre los átomos de carbono se describen como insaturados. Es probable que haya oído hablar de las grasas insaturadas. Son moléculas orgánicas complejas con largas cadenas de átomos de carbono, que contienen al menos un doble enlace entre átomos de carbono. Las moléculas de hidrocarburos insaturados que contienen uno o más dobles enlaces se denominan alquenos. Los átomos de carbono con doble enlace están unidos por dos enlaces: σ y π. Los dobles y triples enlaces dan lugar a una geometría distinta alrededor del átomo de carbono que participa en ellos, lo que origina diferencias importantes en la forma y las propiedades moleculares. Las distintas geometrías son la causa de las diferentes propiedades de las grasas insaturadas frente a las saturadas.
El eteno, C2H4, es el alqueno más simple. Cada átomo de carbono del eteno, denominado comúnmente etileno, tiene una estructura planar trigonal. El segundo integrante de la serie es el propeno (propileno) (); los isómeros del buteno le siguen en la serie. Cuatro átomos de carbono en la cadena del buteno permiten la formación de isómeros basados en la posición del doble enlace, así como una nueva forma de isomerismo.
El etileno (nombre industrial común del eteno) es una materia prima básica en la producción de polietileno y otros compuestos importantes. En 2010 se produjeron más de 135 millones de toneladas de etileno en todo el mundo para su uso en las industrias de polímeros, petroquímica y plásticos. El etileno se produce industrialmente en un proceso llamado craqueo, en el que las largas cadenas de hidrocarburos de una mezcla de petróleo se rompen en moléculas más pequeñas.
El nombre de un alqueno se deriva del nombre del alcano con el mismo número de átomos de carbono. La presencia del doble enlace se indica al sustituir el sufijo -ano por el sufijo -eno. La ubicación del doble enlace se identifica al designar el menor de los números de los átomos de carbono que participan en el doble enlace:
### Isómeros de alquenos
Las moléculas de 1-buteno y 2-buteno son isómeros estructurales; la disposición de los átomos en estas dos moléculas difiere. Como ejemplo de las diferencias de disposición, el primer átomo de carbono del 1-buteno está enlazado a dos átomos de hidrógeno; el primer átomo de carbono del 2-buteno está enlazado a tres átomos de hidrógeno.
El compuesto 2-buteno y algunos otros alquenos también forman un segundo tipo de isómero llamado isómero geométrico. En un conjunto de isómeros geométricos, los mismos tipos de átomos están unidos entre sí en el mismo orden, pero las geometrías de las dos moléculas difieren. Los isómeros geométricos de los alquenos difieren en la orientación de los grupos a ambos lados de un enlace .
Los átomos de carbono rotan libremente alrededor de un enlace simple, pero no alrededor de un doble enlace; el doble enlace es rígido. Esto permite tener dos isómeros del 2-buteno: uno con ambos grupos de metilo en el mismo lado del doble enlace y el otro con los grupos de metilo en lados opuestos. Cuando se dibujan estructuras de buteno con ángulos de enlace de 120° alrededor de los átomos de carbono hibridados sp2 que participan en el doble enlace, los isómeros son evidentes. El isómero de 2-buteno en el que los dos grupos metilo están en el mismo lado se denomina isómero cis; el que tiene los dos grupos de metilo en lados opuestos se denomina isómero trans (). Las diferentes geometrías producen distintas propiedades físicas, como el punto de ebullición, que posibilitan la separación de los isómeros:
Los alquenos son mucho más reactivos que los alcanos porque la fracción es un grupo funcional reactivo. El enlace π, al ser más débil, se altera mucho más fácilmente que el enlace σ. Por lo tanto, los alquenos sufren una reacción característica en la que el enlace π se rompe y se sustituye por dos enlaces σ. Esta reacción se llama reacción de adición. La hibridación de los átomos de carbono en el doble enlace de un alqueno cambia de sp2 a sp3 durante una reacción de adición. Por ejemplo, los halógenos se suman al doble enlace en un alqueno en lugar de sustituir al hidrógeno, como ocurre en un alcano:
### Alquinos
Las moléculas de hidrocarburos con uno o más triples enlaces se denominan alquinos; constituyen otra serie de hidrocarburos insaturados. Dos átomos de carbono unidos por un triple enlace están unidos por un enlace σ y dos enlaces π. Los carbonos sp-hibridados que intervienen en el triple enlace tienen ángulos de enlace de 180°, lo que da a este tipo de enlaces una forma lineal, parecida a una varilla.
El miembro más simple de la serie de alquinos es el etileno, C2H2, comúnmente llamado acetileno. La estructura de Lewis para el etileno, una molécula lineal, es:
La nomenclatura de la IUPAC para los alquinos es similar a la de los alquenos, salvo que el sufijo -ino se utiliza para indicar un triple enlace en la cadena. Por ejemplo, se denomina 1-butino.
Químicamente, los alquinos se parecen a los alquenos. Dado que el grupo funcional tiene dos enlaces π, los alquinos suelen reaccionar aún más fácilmente y reaccionan con el doble de reactivo en las reacciones de adición. La reacción del acetileno con el bromo es un ejemplo típico:
El acetileno y los demás alquinos también arden con facilidad. Un soplete de acetileno aprovecha el alto calor de combustión del acetileno.
### Hidrocarburos aromáticos
El benceno, C6H6, es el miembro más simple de una gran familia de hidrocarburos, llamados hidrocarburos aromáticos. Estos compuestos contienen estructuras de anillo y presentan enlaces que deben describirse con el concepto de híbrido de resonancia de la teoría del enlace de valencia o el concepto de deslocalización de la teoría de orbitales moleculares. (Para repasar estos conceptos, consulte los capítulos anteriores sobre el enlace químico). Las estructuras de resonancia del benceno, C6H6, son:
La teoría del enlace de valencia describe la molécula de benceno y otras moléculas planas de hidrocarburos aromáticos como anillos hexagonales de átomos de carbono hibridados sp2 con el orbital p no hibridado de cada átomo de carbono perpendicular al plano del anillo. Tres electrones de valencia en los orbitales híbridos sp2 de cada átomo de carbono y el electrón de valencia de cada átomo de hidrógeno forman el marco de los enlaces σ en la molécula de benceno. El cuarto electrón de valencia de cada átomo de carbono se comparte con un átomo de carbono adyacente en sus orbitales p no hibridados para dar lugar a los enlaces π. Sin embargo, el benceno no presenta las características típicas de un alqueno. Cada uno de los seis enlaces entre sus átomos de carbono es equivalente y presenta propiedades intermedias entre las de un enlace simple C-C y un doble enlace . Para representar este enlace único, las fórmulas estructurales del benceno y sus derivados se dibujan con enlaces simples entre los átomos de carbono y un círculo dentro del anillo, como se muestra en la .
Hay muchos derivados del benceno. Los átomos de hidrógeno pueden sustituirse por muchos sustituyentes diferentes. Los compuestos aromáticos experimentan más fácilmente reacciones de sustitución que de adición; la sustitución de uno de los átomos de hidrógeno por otro sustituyente dejará intactos los dobles enlaces deslocalizados. Los siguientes son ejemplos típicos de derivados bencénicos sustituidos:
El tolueno y el xileno son importantes disolventes y materias primas en la industria química. El estireno se utiliza para producir el polímero poliestireno.
### Conceptos clave y resumen
Los enlaces fuertes y estables entre los átomos de carbono producen moléculas complejas que contienen cadenas, ramas y anillos. La química de estos compuestos se denomina química orgánica. Los hidrocarburos son compuestos orgánicos formados únicamente por carbono e hidrógeno. Los alcanos son hidrocarburos saturados, es decir, hidrocarburos que solo contienen enlaces simples. Los alquenos contienen uno o más dobles enlaces carbono-carbono. Los alquinos contienen uno o más triples enlaces carbono-carbono. Los hidrocarburos aromáticos contienen estructuras de anillo con sistemas de electrones π deslocalizados.
### Ejercicios de química del final del capítulo
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# Química orgánica
## Alcoholes y éteres
En esta sección, aprenderemos sobre los alcoholes y los éteres.
### Alcoholes
La incorporación de un átomo de oxígeno en moléculas que contienen carbono e hidrógeno da lugar a nuevos grupos funcionales y nuevas familias de compuestos. Cuando el átomo de oxígeno está unido por enlaces simples, la molécula es un alcohol o un éter.
Los alcoholes son derivados de los hidrocarburos en los que un grupo –OH sustituye a un átomo de hidrógeno. Aunque todos los alcoholes tienen uno o más grupos funcionales hidroxilo (–OH), no se comportan como las bases, como el NaOH y el KOH. El NaOH y el KOH son compuestos iónicos que contienen iones de OH-. Los alcoholes son moléculas covalentes; el grupo –OH de una molécula de alcohol está unido a un átomo de carbono mediante un enlace covalente.
El etanol, CH3CH2OH, también llamado alcohol etílico, es un alcohol especialmente importante para el uso humano. El etanol es el alcohol producido por algunas especies de levadura que se encuentra en el vino, la cerveza y las bebidas destiladas. Desde hace tiempo, el ser humano lo prepara aprovechando los esfuerzos metabólicos de las levaduras en la fermentación de diversos azúcares.
Se sintetizan grandes cantidades de etanol a partir de la reacción de adición de agua con etileno utilizando un ácido como catalizador.
Se pueden fabricar alcoholes que contengan dos o más grupos hidroxilos. Algunos ejemplos son el 1,2-etanodiol (etilenglicol, utilizado en los anticongelantes) y el 1,2,3-propanetriol (glicerina, utilizada como disolvente en cosméticos y medicamentos):
### Designación de los alcoholes
El nombre de un alcohol proviene del hidrocarburo del que deriva. La -e final en el nombre del hidrocarburo se sustituye por -ol, y el átomo de carbono al que se une el grupo–OH se indica con un número que se coloca antes del nombre.La IUPAC adoptó nuevas directrices de nomenclatura en 2013 que exigen que este número se coloque como un "infijo" en lugar de un prefijo. Por ejemplo, el nuevo nombre del 2-propanol sería propan-2-ol. La adopción generalizada de esta nueva nomenclatura llevará algún tiempo, por lo que se recomienda a los estudiantes que se familiaricen con los protocolos de denominación antiguos y nuevos.
### Éteres
Los éteres son compuestos que contienen el grupo funcional –O–. Los éteres no tienen ningún sufijo designado como los demás tipos de moléculas que hemos nombrado hasta ahora. En el sistema de la IUPAC, el átomo de oxígeno y la rama de carbono más pequeña se denominan sustituyente alcoxi y el resto de la molécula, cadena base, como en los alcanos. Como se indica en el siguiente compuesto, los símbolos rojos representan el grupo alquilo menor y el átomo de oxígeno, que se denominaría "metoxi". La rama de carbono más grande sería el etano, con lo que la molécula sería metoxietano. Muchos éteres se designan con nombres comunes en lugar de los del sistema de la IUPAC. Para los nombres comunes, las dos ramas conectadas al átomo de oxígeno se designan por separado y van seguidas de "éter". El nombre común del compuesto que aparece en el es etilmetil éter:
Los éteres se obtienen a partir de alcoholes mediante la eliminación de una molécula de agua a partir de dos moléculas del alcohol. Por ejemplo, cuando el etanol se trata con una cantidad limitada de ácido sulfúrico y se calienta a 140 °C, se forma éter dietílico y agua:
En la fórmula general de los éteres, R—O—R, los grupos hidrocarburos (R) pueden ser iguales o diferentes. El éter dietílico, el compuesto más utilizado de esta clase, es un líquido incoloro, volátil y muy inflamable. Se utilizó por primera vez en 1846 como anestésico, aunque ahora hay mejores anestésicos que han ocupado su lugar. El éter dietílico y otros éteres se utilizan en la actualidad principalmente como disolventes de gomas, grasas, ceras y resinas. El metil tert-butil éter, C4H9OCH3 (abreviado MTBE [las partes en cursiva de los nombres no se tienen en cuenta a la hora de clasificar los grupos alfabéticamente], por lo que el butilo viene después del metilo en el nombre común), se utiliza como aditivo para la gasolina. El MTBE pertenece a un grupo de productos químicos conocidos como oxigenados por su capacidad de aumentar el contenido de oxígeno de la gasolina.
### Conceptos clave y resumen
Muchos compuestos orgánicos que no son hidrocarburos pueden considerarse derivados. Un derivado de hidrocarburo se forma al sustituir uno o más átomos de hidrógeno de un hidrocarburo por un grupo funcional que contenga al menos un átomo de un elemento distinto al carbono o al hidrógeno. Las propiedades de los derivados de los hidrocarburos se determinan en gran medida por el grupo funcional. El grupo –OH es el grupo funcional de un alcohol. El grupo –R–O–R– es el grupo funcional de un éter.
### Ejercicios de química del final del capítulo
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# Química orgánica
## Aldehídos, cetonas, ácidos carboxílicos y ésteres
Otra clase de moléculas orgánicas contiene un átomo de carbono conectado a un átomo de oxígeno mediante un doble enlace, comúnmente llamado grupo carbonilo. El carbono trigonal plano del grupo carbonilo se une a otros dos sustituyentes para originar varias subfamilias (aldehídos, cetonas, ácidos carboxílicos y ésteres), las cuales se describen en esta sección.
### Aldehídos y cetonas
Tanto los aldehídos como las cetonas contienen un grupo carbonilo: un grupo funcional con doble enlace carbono-oxígeno. Los nombres de los compuestos aldehídicos y cetónicos se derivan mediante reglas de nomenclatura similares a las de los alcanos y alcoholes, y constan de los sufijos identificadores de clase -al y -ona, respectivamente.
En un aldehído, el grupo carbonilo está enlazado al menos a un átomo de hidrógeno. En una cetona, el grupo carbonilo está enlazado a dos átomos de carbono.
Como texto, un grupo aldehído se representa como –CHO; la cetona se representa como –C(O)– o –CO–.
Tanto en los aldehídos como en las cetonas, la geometría alrededor del átomo de carbono del grupo carbonilo es trigonal plana; el átomo de carbono presenta hibridación sp2. Dos de los orbitales sp2 en el átomo de carbono del grupo carbonilo se utilizan para formar enlaces σ con los demás átomos de carbono o hidrógeno de una molécula. El orbital híbrido sp2 remanente forma un enlace σ con el átomo de oxígeno. El orbital p no hibridado del átomo de carbono en el grupo carbonilo se superpone con un orbital p del átomo de oxígeno para formar el enlace π en el doble enlace.
Al igual que el enlace en el dióxido de carbono, el enlace de un grupo carbonilo es polar (recordemos que el oxígeno es significativamente más electronegativo que el carbono, y los electrones compartidos son atraídos hacia el átomo de oxígeno y alejados del átomo de carbono). Muchas de las reacciones de aldehídos y cetonas comienzan con la reacción entre una base de Lewis y el átomo de carbono del extremo positivo del enlace polar para dar lugar a un intermedio inestable que posteriormente sufre uno o más reajustes estructurales para formar el producto final ().
La importancia de la estructura molecular en la reactividad de los compuestos orgánicos se ilustra con las reacciones que producen aldehídos y cetonas. Podemos preparar un grupo carbonilo mediante la oxidación de un alcohol: en las moléculas orgánicas, se dice que la oxidación de un átomo de carbono se produce cuando un enlace carbono-hidrógeno se sustituye por un enlace carbono-oxígeno. La reacción inversa (la sustitución de un enlace carbono-oxígeno por un enlace carbono-hidrógeno) es la reducción de ese átomo de carbono. Recordemos que al oxígeno se le asigna un número de oxidación -2, a menos que sea elemental o esté unido a un flúor. Al hidrógeno se le asigna un número de oxidación de +1, a menos que esté unido a un metal. Dado que el carbono no tiene ninguna regla específica, su número de oxidación se determina algebraicamente mediante la factorización de los átomos a los que está unido y la carga global de la molécula o del ion. En general, un átomo de carbono unido a uno de oxígeno tendrá un número de oxidación más positivo y un átomo de carbono unido a uno de hidrógeno tendrá un número de oxidación más negativo. Esto debería encajar a la perfección con la comprensión que se tenga acerca de la polaridad de los enlaces C–O y C–H. Los otros reactivos y los posibles productos de estas reacciones están fuera del alcance de este capítulo, por lo que nos centraremos apenas en los cambios en los átomos de carbono.
Los aldehídos se preparan habitualmente mediante la oxidación de alcoholes, cuyo grupo funcional –OH se sitúa en el átomo de carbono al final de la cadena de átomos de carbono del alcohol:
Los alcoholes que tienen sus grupos –OH en el centro de la cadena son necesarios para sintetizar una cetona, lo cual exige que el grupo carbonilo esté enlazado a otros dos átomos de carbono.
Un alcohol con su grupo –OH enlazado a un átomo de carbono que no está enlazado a ningún otro átomo de carbono formará un aldehído. Un alcohol con su grupo –OH enlazado a otros dos átomos de carbono formará una cetona. Si hay tres carbonos unidos al carbono enlazado al –OH, la molécula no tendrá ningún enlace C-H que sustituir, por lo que no será susceptible de oxidación.
El formaldehído, un aldehído de fórmula HCHO, es un gas incoloro de olor penetrante e irritante. Se vende en una solución acuosa que recibe el nombre de formalina, la cual contiene aproximadamente 37 % de formaldehído en peso. El formaldehído provoca la coagulación de las proteínas, por lo que mata las bacterias (y cualquier otro organismo vivo) y detiene muchos de los procesos biológicos que causan la descomposición de los tejidos. Así, el formaldehído se utiliza para conservar muestras de tejidos y embalsamar cadáveres. También se utiliza para esterilizar la tierra u otros materiales. El formaldehído se emplea en la fabricación de baquelita, un plástico duro de gran resistencia química y eléctrica.
La dimetil cetona, CH3COCH3, mejor conocida como acetona, es la cetona más simple. Se fabrica comercialmente mediante la fermentación de maíz o melaza, o por oxidación del 2-propanol. La acetona es un líquido incoloro. Entre sus múltiples usos se encuentran como disolvente de lacas (incluido el esmalte de uñas), acetato de celulosa, nitrato de celulosa, acetileno, plásticos y barnices; como removedor de pinturas y barnices, y como disolvente en la fabricación de productos farmacéuticos y químicos.
### Ácidos carboxílicos y ésteres
El olor del vinagre se debe a la presencia de ácido acético, un ácido carboxílico, en el vinagre. El olor de los plátanos maduros y de muchas otras frutas se debe a la presencia de ésteres, compuestos que se pueden preparar por la reacción de un ácido carboxílico con un alcohol. Dado que los ésteres no tienen enlaces de hidrógeno entre las moléculas, tienen presiones de vapor más bajas que los alcoholes y los ácidos carboxílicos de los que derivan (vea la ).
Tanto los ácidos carboxílicos como los ésteres contienen un grupo carbonilo con un segundo átomo de oxígeno unido al átomo de carbono del grupo carbonilo mediante un enlace simple. En el ácido carboxílico, el segundo átomo de oxígeno también se une a un átomo de hidrógeno. En un éster, el segundo átomo de oxígeno se une a otro átomo de carbono. Los nombres de los ácidos carboxílicos y los ésteres contienen prefijos que denotan las longitudes de las cadenas de carbono en las moléculas y se derivan conforme a reglas de nomenclatura semejantes a las de los ácidos inorgánicos y las sales (vea estos ejemplos).
Los grupos funcionales para un ácido y para un éster se muestran en rojo en estas fórmulas.
El átomo de hidrógeno del grupo funcional de un ácido carboxílico reacciona con una base para formar una sal iónica.
Los ácidos carboxílicos son débiles (vea el capítulo sobre ácidos y bases), lo que significa que no se ionizan al 100 % en el agua. Por lo general, apenas un 1 % de las moléculas de un ácido carboxílico disuelto en agua se ionizan en un momento dado. Las moléculas restantes no se disocian en la solución.
Los ácidos carboxílicos se preparan mediante la oxidación de aldehídos o alcoholes, cuyo grupo funcional –OH está situado en el átomo de carbono del final de la cadena de átomos de carbono del alcohol.
Los ésteres se producen por la reacción de los ácidos con los alcoholes. Por ejemplo, el éster acetato de etilo, CH3CO2CH2CH3, se forma cuando el ácido acético reacciona con el etanol:
El ácido carboxílico más simple es el ácido fórmico, HCO2H, conocido desde 1670. Su nombre proviene de la palabra latina formicus, que significa "hormiga"; se aisló por primera vez mediante la destilación de hormigas rojas. Es responsable en parte del dolor y de la irritación que causan las picaduras de hormigas y avispas, y da ese olor característico que a veces se detecta en los hormigueros.
El ácido acético, CH3CO2H, constituye del 3 % al 6 % del vinagre. El vinagre de sidra se produce con la fermentación del zumo de manzana sin la presencia de oxígeno. Las células de levadura presentes en el zumo llevan a cabo las reacciones de fermentación. Las reacciones de fermentación transforman el azúcar presente en el zumo en etanol y luego en ácido acético. El ácido acético puro tiene un olor penetrante y produce quemaduras dolorosas. Es un excelente disolvente para muchos compuestos orgánicos y algunos inorgánicos, y es esencial en la producción de acetato de celulosa, un componente de muchas fibras sintéticas como el rayón.
Los olores y sabores distintivos y atractivos de muchas flores, perfumes y frutas maduras se deben a la presencia de uno o más ésteres (). Entre los ésteres naturales más importantes se encuentran las grasas (como la manteca de cerdo, el sebo y la mantequilla) y los aceites (como el de linaza, el de algodón y el de oliva), que son ésteres del alcohol glicerol trihidroxilo C3H5(OH)3, con grandes ácidos carboxílicos, tales como el ácido palmítico, CH3(CH2)14CO2H, el ácido esteárico, CH3(CH2)16CO2H, y el ácido oleico, El ácido oleico es un ácido insaturado; contiene un doble enlace . Los ácidos palmítico y esteárico son ácidos saturados que no contienen ni dobles ni triples enlaces.
### Conceptos clave y resumen
Los grupos funcionales relacionados con el grupo carbonilo comprenden el grupo –CHO de un aldehído, el grupo –CO– de una cetona, el grupo–CO2H de un ácido carboxílico y el grupo –CO2R de un éster. El grupo carbonilo, un doble enlace carbono-oxígeno, es la estructura clave en estas clases de moléculas orgánicas: Los aldehídos contienen al menos un átomo de hidrógeno unido al átomo de carbono del carbonilo; las cetonas contienen dos grupos de carbono unidos al átomo de carbono del carbonilo; los ácidos carboxílicos contienen un grupo hidroxilo unido al átomo de carbono del carbonilo, y los ésteres contienen un átomo de oxígeno unido a otro grupo de carbono conectado al átomo de carbono del carbonilo. Todos estos compuestos contienen átomos de carbono oxidados en relación con el átomo de carbono de un grupo de alcohol.
### Ejercicios de química del final del capítulo
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# Química orgánica
## Aminas y amidas
Las aminas son moléculas que contienen enlaces carbono-nitrógeno. El átomo de nitrógeno de una amina tiene un par solitario de electrones y tres enlaces con otros átomos, ya sean de carbono o de hidrógeno. Se utilizan varias nomenclaturas para derivar los nombres de las aminas, pero todas implican el sufijo identificador de clase -ina, como se ilustra aquí en algunos ejemplos sencillos:
En algunas aminas, el átomo de nitrógeno sustituye a un átomo de carbono en un hidrocarburo aromático. La piridina () es una de estas aminas heterocíclicas. Un compuesto heterocíclico contiene átomos de dos o más elementos diferentes en su estructura de anillo.
Al igual que el amoníaco, las aminas son bases débiles debido al par solitario de electrones de sus átomos de nitrógeno.
La alcalinidad del átomo de nitrógeno de una amina desempeña una función importante en gran parte de la química del compuesto. Los grupos funcionales de las aminas se encuentran en una gran variedad de compuestos, como tintes naturales y sintéticos, polímeros, vitaminas y medicamentos como la penicilina y la codeína. También se encuentran en muchas moléculas esenciales para la vida, como los aminoácidos, las hormonas, los neurotransmisores y el ADN.
Las amidas son moléculas que contienen átomos de nitrógeno conectados al átomo de carbono de un grupo carbonilo. Al igual que las aminas, se pueden utilizar varias reglas de nomenclatura para designar las amidas, pero todas incluyen el uso del sufijo específico de la clase -amida:
Las amidas se producen cuando los ácidos carboxílicos reaccionan con aminas o amoníaco en un proceso que recibe el nombre de amidación. Se elimina una molécula de agua de la reacción, y la amida se forma a partir de los trozos restantes del ácido carboxílico y la amina (observe la similitud con la formación de un éster a partir de un ácido carboxílico y un alcohol que se analizó en la sección anterior).
La reacción entre las aminas y los ácidos carboxílicos para formar amidas es importante biológicamente. Es a través de esta reacción que los aminoácidos (moléculas que contienen sustituciones de amina y ácido carboxílico) se unen en un polímero para formar proteínas.
En esta tabla se resumen las estructuras analizadas en este capítulo.
### Conceptos clave y resumen
La adición de nitrógeno en un marco orgánico da lugar a dos familias de moléculas. Los compuestos que contienen un átomo de nitrógeno enlazado en una estructura de hidrocarburo se clasifican como aminas. Los compuestos que tienen un átomo de nitrógeno enlazado a un lado de un grupo carbonilo se clasifican como amidas. Las aminas son un grupo funcional básico. Las aminas y los ácidos carboxílicos se combinan en una reacción de condensación para formar amidas.
### Ejercicios de química del final del capítulo
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# Foundations
## Introduction
Just like a building needs a firm foundation to support it, your study of algebra needs to have a firm foundation. To ensure this, we begin this book with a review of arithmetic operations with whole numbers, integers, fractions, and decimals, so that you have a solid base that will support your study of algebra. |
# Foundations
## Introduction to Whole Numbers
As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.
### Use Place Value with Whole Numbers
The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.
Counting Numbers: 1, 2, 3, …
Whole Numbers: 0, 1, 2, 3, …
The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly.
We can visualize counting numbers and whole numbers on a number line (see ).
Our number system is called a place value system, because the value of a digit depends on its position in a number. shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.
When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see ). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.
We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.
In 2013, the U.S. Census Bureau estimated the population of the state of New York as 19,651,127. We could say the population of New York was approximately 20 million. In many cases, you don’t need the exact value; an approximate number is good enough.
The process of approximating a number is called rounding. Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of New York is approximately 20 million means that we rounded to the millions place.
In algebra, we use a letter of the alphabet to represent a number whose value may change or is unknown. Commonly used symbols are a, b, c, m, n, x, and y. Further discussion of constants and variables appears later in this section.
### Identify Multiples and Apply Divisibility Tests
The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.
Similarly, a multiple of 3 would be the product of a counting number and 3.
We could find the multiples of any number by continuing this process.
shows the multiples of 2 through 9 for the first 12 counting numbers.
Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 15 by 3, we get a counting number. In fact, is 5, so 15 is
Look at the multiples of 5 in . They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:
### Find Prime Factorizations and Least Common Multiples
In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.
Since we say that 8 and 9 are factors of 72. When we write we say we have factored 72.
Other ways to factor 72 are Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.
Some numbers, like 72, have many factors. Other numbers have only two factors.
The counting numbers from 2 to 19 are listed in , with their factors. Make sure to agree with the “prime” or “composite” label for each!
The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2.
A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.
To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime!
If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.
One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators. Two methods are used most often to find the least common multiple and we will look at both of them.
The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:
Notice that some numbers appear in both lists. They are the common multiples of 12 and 18.
We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.
The procedure box lists the steps to take to find the LCM using the prime factors method we used above for 12 and 18.
Our second method to find the least common multiple of two numbers is to use The Prime Factors Method. Let’s find the LCM of 12 and 18 again, this time using their prime factors.
Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM So 36 is the least common multiple of 12 and 18.
By matching up the common primes, each common prime factor is used only once. This way you are sure that 36 is the least common multiple.
### Key Concepts
1. Place Value as in .
2. Name a Whole Number in Words
3. Write a Whole Number Using Digits
4. Round Whole Numbers
5. Divisibility Tests: A number is divisible by:
6. Find the Prime Factorization of a Composite Number
7. Find the Least Common Multiple by Listing Multiples
8. Find the Least Common Multiple Using the Prime Factors Method
### Practice Makes Perfect
Use Place Value with Whole Numbers
In the following exercises, find the place value of each digit in the given numbers.
In the following exercises, name each number using words.
In the following exercises, write each number as a whole number using digits.
In the following, round to the indicated place value.
In the following exercises, round each number to the nearest ⓐ hundred, ⓑ thousand, ⓒ ten thousand.
Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, 3, 5, 6, and 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of the each pair of numbers using the multiples method.
In the following exercises, find the least common multiple of each pair of numbers using the prime factors method.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Foundations
## Use the Language of Algebra
### Use Variables and Algebraic Symbols
Suppose this year Greg is 20 years old and Alex is 23. You know that Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right? In the language of algebra, we say that Greg’s age and Alex’s age are variables and the 3 is a constant. The ages change (“vary”) but the 3 years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by 3 years, 3 is the constant.
In algebra, we use letters of the alphabet to represent variables. So if we call Greg’s age g, then we could use to represent Alex’s age. See .
The letters used to represent these changing ages are called variables. The letters most commonly used for variables are x, y, a, b, and c.
To write algebraically, we need some operation symbols as well as numbers and variables. There are several types of symbols we will be using.
There are four basic arithmetic operations: addition, subtraction, multiplication, and division. We’ll list the symbols used to indicate these operations in the table below. You’ll probably recognize some of them.
We perform these operations on two numbers. When translating from symbolic form to English, or from English to symbolic form, pay attention to the words “of” and “and.”
1. The difference of 9 and 2 means subtract 9 and 2, in other words, 9 minus 2, which we write symbolically as
2. The product of 4 and 8 means multiply 4 and 8, in other words 4 times 8, which we write symbolically as
In algebra, the cross symbol, is not used to show multiplication because that symbol may cause confusion. Does 3xy mean (‘three times y’) or (three times x times y)? To make it clear, use or parentheses for multiplication.
When two quantities have the same value, we say they are equal and connect them with an equal sign.
On the number line, the numbers get larger as they go from left to right. The number line can be used to explain the symbols “<” and “>.”
The expressions a < b or a > b can be read from left to right or right to left, though in English we usually read from left to right (). In general, a < b is equivalent to b > a. For example 7 < 11 is equivalent to 11 > 7. And a > b is equivalent to b < a. For example 17 > 4 is equivalent to 4 < 17.
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help to make clear which expressions are to be kept together and separate from other expressions. We will introduce three types now.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
What is the difference in English between a phrase and a sentence? A phrase expresses a single thought that is incomplete by itself, but a sentence makes a complete statement. “Running very fast” is a phrase, but “The football player was running very fast” is a sentence. A sentence has a subject and a verb. In algebra, we have expressions and equations.
An expression is like an English phrase. Here are some examples of expressions:
Notice that the English phrases do not form a complete sentence because the phrase does not have a verb.
An equation is two expressions linked with an equal sign. When you read the words the symbols represent in an equation, you have a complete sentence in English. The equal sign gives the verb.
Here are some examples of equations.
Suppose we need to multiply 2 nine times. We could write this as This is tedious and it can be hard to keep track of all those 2s, so we use exponents. We write as and as In expressions such as the 2 is called the base and the 3 is called the exponent. The exponent tells us how many times we need to multiply the base.
We read as “two to the third power” or “two cubed.”
We say is in exponential notation and is in expanded notation.
While we read as “a to the power,” we usually read:
1. “a squared”
2. “a cubed”
We’ll see later why and have special names.
shows how we read some expressions with exponents.
### Simplify Expressions Using the Order of Operations
To simplify an expression means to do all the math possible. For example, to simplify we’d first multiply to get 8 and then add the 1 to get 9. A good habit to develop is to work down the page, writing each step of the process below the previous step. The example just described would look like this:
By not using an equal sign when you simplify an expression, you may avoid confusing expressions with equations.
We’ve introduced most of the symbols and notation used in algebra, but now we need to clarify the order of operations. Otherwise, expressions may have different meanings, and they may result in different values. For example, consider the expression:
If you simplify this expression, what do you get?
Some students say 49,
Others say 25,
Imagine the confusion in our banking system if every problem had several different correct answers!
The same expression should give the same result. So mathematicians early on established some guidelines that are called the Order of Operations.
Students often ask, “How will I remember the order?” Here is a way to help you remember: Take the first letter of each key word and substitute the silly phrase: “Please Excuse My Dear Aunt Sally.”
It’s good that “My Dear” goes together, as this reminds us that multiplication and division have equal priority. We do not always do multiplication before division or always do division before multiplication. We do them in order from left to right.
Similarly, “Aunt Sally” goes together and so reminds us that addition and subtraction also have equal priority and we do them in order from left to right.
Let’s try an example.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
### Evaluate an Expression
In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To evaluate an expression means to find the value of the expression when the variable is replaced by a given number.
To evaluate an expression, substitute that number for the variable in the expression and then simplify the expression.
### Identify and Combine Like Terms
Algebraic expressions are made up of terms. A term is a constant, or the product of a constant and one or more variables.
Examples of terms are
The constant that multiplies the variable is called the coefficient.
Think of the coefficient as the number in front of the variable. The coefficient of the term 3x is 3. When we write x, the coefficient is 1, since
Some terms share common traits. Look at the following 6 terms. Which ones seem to have traits in common?
The 7 and the 4 are both constant terms.
The 5x and the 3x are both terms with x.
The and the are both terms with
When two terms are constants or have the same variable and exponent, we say they are like terms.
1. 7 and 4 are like terms.
2. 5x and 3x are like terms.
3. and are like terms.
Adding or subtracting terms forms an expression. In the expression from , the three terms are and 8.
If there are like terms in an expression, you can simplify the expression by combining the like terms. What do you think would simplify to? If you thought 12x, you would be right!
Add the coefficients and keep the same variable. It doesn’t matter what x is—if you have 4 of something and add 7 more of the same thing and then add 1 more, the result is 12 of them. For example, 4 oranges plus 7 oranges plus 1 orange is 12 oranges. We will discuss the mathematical properties behind this later.
Simplify:
Add the coefficients. 12x
### Translate an English Phrase to an Algebraic Expression
In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. summarizes them.
Look closely at these phrases using the four operations:
Each phrase tells us to operate on two numbers. Look for the words of and and to find the numbers.
How old will you be in eight years? What age is eight more years than your age now? Did you add 8 to your present age? Eight “more than” means 8 added to your present age. How old were you seven years ago? This is 7 years less than your age now. You subtract 7 from your present age. Seven “less than” means 7 subtracted from your present age.
Later in this course, we’ll apply our skills in algebra to solving applications. The first step will be to translate an English phrase to an algebraic expression. We’ll see how to do this in the next two examples.
### Key Concepts
1. Notation The result is…
2. Inequality
3. Inequality Symbols Words
4. Grouping Symbols
5. Exponential Notation
6. Order of Operations: When simplifying mathematical expressions perform the operations in the following order:
7. Combine Like Terms
### Practice Makes Perfect
Use Variables and Algebraic Symbols
In the following exercises, translate from algebra to English.
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
In the following exercises, simplify using the order of operations.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
In the following exercises, identify the terms in each expression.
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the phrases into algebraic expressions.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ Use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Foundations
## Add and Subtract Integers
### Use Negatives and Opposites
Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. Negative numbers are numbers less than The negative numbers are to the left of zero on the number line. See .
The arrows on the ends of the number line indicate that the numbers keep going forever. There is no biggest positive number, and there is no smallest negative number.
Is zero a positive or a negative number? Numbers larger than zero are positive, and numbers smaller than zero are negative. Zero is neither positive nor negative.
Consider how numbers are ordered on the number line. Going from left to right, the numbers increase in value. Going from right to left, the numbers decrease in value. See .
Remember that we use the notation:
a < b (read “a is less than b”) when a is to the left of b on the number line.
a > b (read “a is greater than b”) when a is to the right of b on the number line.
Now we need to extend the number line which showed the whole numbers to include negative numbers, too. The numbers marked by points in are called the integers. The integers are the numbers
You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers 2 and are the same distance from zero, they are called opposites. The opposite of 2 is and the opposite of is 2.
illustrates the definition.
Sometimes in algebra the same symbol has different meanings. Just like some words in English, the specific meaning becomes clear by looking at how it is used. You have seen the symbol “−” used in three different ways.
Our work with opposites gives us a way to define the integers.The whole numbers and their opposites are called the integers. The integers are the numbers
When evaluating the opposite of a variable, we must be very careful. Without knowing whether the variable represents a positive or negative number, we don’t know whether is positive or negative. We can see this in .
### Simplify: Expressions with Absolute Value
We saw that numbers such as are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.
For example,
1. units away from so
2. units away from so
illustrates this idea.
The absolute value of a number is never negative (because distance cannot be negative). The only number with absolute value equal to zero is the number zero itself, because the distance from on the number line is zero units.
Mathematicians say it more precisely, “absolute values are always non-negative.” Non-negative means greater than or equal to zero.
In the next example, we’ll order expressions with absolute values. Remember, positive numbers are always greater than negative numbers!
We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.
In the next example, we simplify the expressions inside absolute value bars first, just as we do with parentheses.
### Add Integers
Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.
We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.
We let one color (blue) represent positive. The other color (red) will represent the negatives. If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.
We will use the counters to show how to add the four addition facts using the numbers and
To add we realize that means the sum of 5 and 3.
Now we will add Watch for similarities to the last example
To add we realize this means the sum of
In what ways were these first two examples similar?
1. The first example adds 5 positives and 3 positives—both positives.
2. The second example adds 5 negatives and 3 negatives—both negatives.
In each case we got 8—either 8 positives or 8 negatives.
When the signs were the same, the counters were all the same color, and so we added them.
So what happens when the signs are different? Let’s add We realize this means the sum of and 3. When the counters were the same color, we put them in a row. When the counters are a different color, we line them up under each other.
Notice that there were more negatives than positives, so the result was negative.
Let’s now add the last combination,
When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.
Now that we have added small positive and negative integers with a model, we can visualize the model in our minds to simplify problems with any numbers.
When you need to add numbers such as you really don’t want to have to count out 37 blue counters and 53 red counters. With the model in your mind, can you visualize what you would do to solve the problem?
Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more red (negative) counters than blue (positive) counters, the sum would be negative. How many more red counters would there be? Because there are 16 more red counters.
Therefore, the sum of is
Let’s try another one. We’ll add Again, imagine 74 red counters and 27 more red counters, so we’d have 101 red counters. This means the sum is
Let’s look again at the results of adding the different combinations of and
Visualize the model as you simplify the expressions in the following examples.
The techniques used up to now extend to more complicated problems, like the ones we’ve seen before. Remember to follow the order of operations!
### Subtract Integers
We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.
Perhaps when you were younger, you read as take away When you use counters, you can think of subtraction the same way!
We will model the four subtraction facts using the numbers and
To subtract we restate the problem as take away
Now we will subtract Watch for similarities to the last example
To subtract we restate this as take away
Notice that these two examples are much alike: The first example, we subtract 3 positives from 5 positives and end up with 2 positives.
In the second example, we subtract 3 negatives from 5 negatives and end up with 2 negatives.
Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.
What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.
1. To subtract we restate it as take away 3.
We start with 5 negatives. We need to take away 3 positives, but we do not have any positives to take away.
Remember, a neutral pair has value zero. If we add 0 to 5 its value is still 5. We add neutral pairs to the 5 negatives until we get 3 positives to take away.
And now, the fourth case, We start with 5 positives. We need to take away 3 negatives, but there are no negatives to take away. So we add neutral pairs until we have 3 negatives to take away.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? In , is the same as and is the same as You will often see this idea, the subtraction property, written as follows:
Look at these two examples.
Of course, when you have a subtraction problem that has only positive numbers, like you just do the subtraction. You already knew how to subtract long ago. But knowing that gives the same answer as helps when you are subtracting negative numbers. Make sure that you understand how and give the same results!
Look at what happens when we subtract a negative.
Subtracting a negative number is like adding a positive!
You will often see this written as
Does that work for other numbers, too? Let’s do the following example and see.
Let’s look again at the results of subtracting the different combinations of and
What happens when there are more than three integers? We just use the order of operations as usual.
### Key Concepts
1. Addition of Positive and Negative Integers
2. Property of Absolute Value: for all numbers. Absolute values are always greater than or equal to zero!
3. Subtraction of Integers
4. Subtraction Property: Subtracting a number is the same as adding its opposite.
### Practice Makes Perfect
Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
In the following exercises, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, evaluate.
Simplify Expressions with Absolute Value
In the following exercises, simplify.
In the following exercises, fill in <, >, or for each of the following pairs of numbers.
In the following exercises, simplify.
In the following exercises, evaluate.
Add Integers
In the following exercises, simplify each expression.
Subtract Integers
In the following exercises, simplify.
In the following exercises, simplify each expression.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Foundations
## Multiply and Divide Integers
### Multiply Integers
Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.
We remember that means add a, b times.
The next two examples are more interesting.
What does it mean to multiply 5 by It means subtract 5, 3 times. Looking at subtraction as “taking away,” it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace. Then we take away 5 three times.
In summary:
Notice that for multiplication of two signed numbers, when the:
1. signs are the same, the product is positive.
2. signs are different, the product is negative.
We’ll put this all together in the chart below.
When we multiply a number by 1, the result is the same number. What happens when we multiply a number by Let’s multiply a positive number and then a negative number by to see what we get.
Each time we multiply a number by we get its opposite!
### Divide Integers
What about division? Division is the inverse operation of multiplication. So, because In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.
Division follows the same rules as multiplication!
For division of two signed numbers, when the:
1. signs are the same, the quotient is positive.
2. signs are different, the quotient is negative.
And remember that we can always check the answer of a division problem by multiplying.
### Simplify Expressions with Integers
What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?
Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.
The next example reminds us to simplify inside parentheses first.
### Evaluate Variable Expressions with Integers
Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.
### Translate Phrases to Expressions with Integers
Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.
When we first introduced the operation symbols, we saw that the expression may be read in several ways. They are listed in the chart below.
Be careful to get a and b in the right order!
Once again, our prior work translating English to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is “product” and for division is “quotient.”
### Use Integers in Applications
We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.
### Key Concepts
1. Multiplication and Division of Two Signed Numbers
2. Strategy for Applications
### Practice Makes Perfect
Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Foundations
## Visualize Fractions
### Find Equivalent Fractions
Fractions are a way to represent parts of a whole. The fraction means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See . The fraction represents two of three equal parts. In the fraction the 2 is called the numerator and the 3 is called the denominator.
If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate pieces, or, in other words, one whole pie.
So This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.
If a pie was cut in pieces and we ate all 6, we ate pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate pieces, or one whole pie. We ate the same amount—one whole pie.
The fractions and have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.
Let’s think of pizzas this time. shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that is equivalent to In other words, they are equivalent fractions.
How can we use mathematics to change into How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into pieces instead of just 2. Mathematically, what we’ve described could be written like this as See .
This model leads to the following property:
If we had cut the pizza differently, we could get
So, we say are equivalent fractions.
### Simplify Fractions
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
1. is simplified because there are no common factors of 2 and 3.
2. is not simplified because is a common factor of 10 and 15.
The phrase reduce a fraction means to simplify the fraction. We simplify, or reduce, a fraction by removing the common factors of the numerator and denominator. A fraction is not simplified until all common factors have been removed. If an expression has fractions, it is not completely simplified until the fractions are simplified.
In , we used the equivalent fractions property to find equivalent fractions. Now we’ll use the equivalent fractions property in reverse to simplify fractions. We can rewrite the property to show both forms together.
Sometimes it may not be easy to find common factors of the numerator and denominator. When this happens, a good idea is to factor the numerator and the denominator into prime numbers. Then divide out the common factors using the equivalent fractions property.
We now summarize the steps you should follow to simplify fractions.
### Multiply Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.
We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with
Now we’ll take of
Notice that now, the whole is divided into 8 equal parts. So
To multiply fractions, we multiply the numerators and multiply the denominators.
When multiplying fractions, the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In , we will multiply negative and a positive, so the product will be negative.
When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a, can be written as So, for example,
### Divide Fractions
Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary.
The reciprocal of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of is
Notice that A number and its reciprocal multiply to 1.
To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.
The reciprocal of is since
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say to be sure we don’t divide by zero!
There are several ways to remember which steps to take to multiply or divide fractions. One way is to repeat the call outs to yourself. If you do this each time you do an exercise, you will have the steps memorized.
1. “To multiply fractions, multiply the numerators and multiply the denominators.”
2. “To divide fractions, multiply the first fraction by the reciprocal of the second.”
Another way is to keep two examples in mind:
The numerators or denominators of some fractions contain fractions themselves. A fraction in which the numerator or the denominator is a fraction is called a complex fraction.
Some examples of complex fractions are:
To simplify a complex fraction, we remember that the fraction bar means division. For example, the complex fraction means
### Simplify Expressions with a Fraction Bar
The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.
To simplify the expression we first simplify the numerator and the denominator separately. Then we divide.
Where does the negative sign go in a fraction? Usually the negative sign is in front of the fraction, but you will sometimes see a fraction with a negative numerator, or sometimes with a negative denominator. Remember that fractions represent division. When the numerator and denominator have different signs, the quotient is negative.
### Translate Phrases to Expressions with Fractions
Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.
The English words quotient and ratio are often used to describe fractions. Remember that “quotient” means division. The quotient of and is the result we get from dividing by or
### Key Concepts
1. Equivalent Fractions Property: If are numbers where then
and
2. Fraction Division: If are numbers where then To divide fractions, multiply the first fraction by the reciprocal of the second.
3. Fraction Multiplication: If are numbers where then To multiply fractions, multiply the numerators and multiply the denominators.
4. Placement of Negative Sign in a Fraction: For any positive numbers
5. Property of One: Any number, except zero, divided by itself is one.
6. Simplify a Fraction
7. Simplify an Expression with a Fraction Bar
### Practice Makes Perfect
Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
Simplify Fractions
In the following exercises, simplify.
Multiply Fractions
In the following exercises, multiply.
Divide Fractions
In the following exercises, divide.
In the following exercises, simplify.
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not? |
# Foundations
## Add and Subtract Fractions
### Add or Subtract Fractions with a Common Denominator
When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.
Now we will do an example that has both addition and subtraction.
### Add or Subtract Fractions with Different Denominators
As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.
After we find the least common denominator of two fractions, we convert the fractions to equivalent fractions with the LCD. Putting these steps together allows us to add and subtract fractions because their denominators will be the same!
When finding the equivalent fractions needed to create the common denominators, there is a quick way to find the number we need to multiply both the numerator and denominator. This method works if we found the LCD by factoring into primes.
Look at the factors of the LCD and then at each column above those factors. The “missing” factors of each denominator are the numbers we need.
In , the LCD, 36, has two factors of 2 and two factors of
The numerator 12 has two factors of 2 but only one of 3—so it is “missing” one 3—we multiply the numerator and denominator by 3.
The numerator 18 is missing one factor of 2—so we multiply the numerator and denominator by 2.
We will apply this method as we subtract the fractions in .
In the next example, one of the fractions has a variable in its numerator. Notice that we do the same steps as when both numerators are numbers.
We now have all four operations for fractions. summarizes fraction operations.
### Use the Order of Operations to Simplify Complex Fractions
We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction by dividing by
Now we’ll look at complex fractions where the numerator or denominator contains an expression that can be simplified. So we first must completely simplify the numerator and denominator separately using the order of operations. Then we divide the numerator by the denominator.
### Evaluate Variable Expressions with Fractions
We have evaluated expressions before, but now we can evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.
The next example will have only variables, no constants.
### Key Concepts
1. Fraction Addition and Subtraction: If are numbers where then
and
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
2. Strategy for Adding or Subtracting Fractions
3. Simplify Complex Fractions
### Practice Makes Perfect
Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Mixed Practice
In the following exercises, simplify.
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
Mixed Practice
In the following exercises, simplify.
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not? |
# Foundations
## Decimals
### Name and Write Decimals
Decimals are another way of writing fractions whose denominators are powers of 10.
Notice that “ten thousand” is a number larger than one, but “one ten-thousandth” is a number smaller than one. The “th” at the end of the name tells you that the number is smaller than one.
When we name a whole number, the name corresponds to the place value based on the powers of ten. We read 10,000 as “ten thousand” and 10,000,000 as “ten million.” Likewise, the names of the decimal places correspond to their fraction values. shows the names of the place values to the left and right of the decimal point.
We summarize the steps needed to name a decimal below.
When we write a check we write both the numerals and the name of the number. Let’s see how to write the decimal from the name.
We summarize the steps to writing a decimal.
### Round Decimals
Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.
We summarize the steps for rounding a decimal here.
### Add and Subtract Decimals
To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.
### Multiply and Divide Decimals
Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first convert them to fractions and then multiply.
So let’s see what we would get as the product of decimals by converting them to fractions first. We will do two examples side-by-side. Look for a pattern!
Notice, in the first example, we multiplied two numbers that each had one digit after the decimal point and the product had two decimal places. In the second example, we multiplied a number with one decimal place by a number with two decimal places and the product had three decimal places.
We multiply the numbers just as we do whole numbers, temporarily ignoring the decimal point. We then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product.
The rules for multiplying positive and negative numbers apply to decimals, too, of course!
When multiplying two numbers,
1. if their signs are the same the product is positive.
2. if their signs are different the product is negative.
When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. Finally, we write the product with the appropriate sign.
In many of your other classes, especially in the sciences, you will multiply decimals by powers of 10 (10, 100, 1000, etc.). If you multiply a few products on paper, you may notice a pattern relating the number of zeros in the power of 10 to number of decimal places we move the decimal point to the right to get the product.
Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.
To divide decimals, determine what power of 10 to multiply the denominator by to make it a whole number. Then multiply the numerator by that same power of Because of the equivalent fractions property, we haven’t changed the value of the fraction! The effect is to move the decimal points in the numerator and denominator the same number of places to the right. For example:
We use the rules for dividing positive and negative numbers with decimals, too. When dividing signed decimals, first determine the sign of the quotient and then divide as if the numbers were both positive. Finally, write the quotient with the appropriate sign.
We review the notation and vocabulary for division:
We’ll write the steps to take when dividing decimals, for easy reference.
A common application of dividing whole numbers into decimals is when we want to find the price of one item that is sold as part of a multi-pack. For example, suppose a case of 24 water bottles costs $3.99. To find the price of one water bottle, we would divide $3.99 by 24. We show this division in . In calculations with money, we will round the answer to the nearest cent (hundredth).
### Convert Decimals, Fractions, and Percents
We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.
Notice, when the number to the left of the decimal is zero, we get a fraction whose numerator is less than its denominator. Fractions like this are called proper fractions.
The steps to take to convert a decimal to a fraction are summarized in the procedure box.
We’ve learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar means division. So can be written or This leads to the following method for converting a fraction to a decimal.
When we divide, we will not always get a zero remainder. Sometimes the quotient ends up with a decimal that repeats. A repeating decimal is a decimal in which the last digit or group of digits repeats endlessly. A bar is placed over the repeating block of digits to indicate it repeats.
A bar is placed over the repeating block of digits to indicate it repeats.
Sometimes we may have to simplify expressions with fractions and decimals together.
A percent is a ratio whose denominator is 100. Percent means per hundred. We use the percent symbol, %, to show percent.
Since a percent is a ratio, it can easily be expressed as a fraction. Percent means per 100, so the denominator of the fraction is 100. We then change the fraction to a decimal by dividing the numerator by the denominator.
Do you see the pattern? To convert a percent number to a decimal number, we move the decimal point two places to the left.
Converting a decimal to a percent makes sense if we remember the definition of percent and keep place value in mind.
To convert a decimal to a percent, remember that percent means per hundred. If we change the decimal to a fraction whose denominator is 100, it is easy to change that fraction to a percent.
Recognize the pattern? To convert a decimal to a percent, we move the decimal point two places to the right and then add the percent sign.
### Key Concepts
1. Name a Decimal
2. Write a Decimal
3. Round a Decimal
4. Add or Subtract Decimals
5. Multiply Decimals
6. Multiply a Decimal by a Power of Ten
7. Divide Decimals
8. Convert a Decimal to a Proper Fraction
9. Convert a Fraction to a Decimal Divide the numerator of the fraction by the denominator.
### Practice Makes Perfect
Name and Write Decimals
In the following exercises, write as a decimal.
In the following exercises, name each decimal.
Round Decimals
In the following exercises, round each number to the nearest tenth.
In the following exercises, round each number to the nearest hundredth.
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide.
Convert Decimals, Fractions and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Foundations
## The Real Numbers
### Simplify Expressions with Square Roots
Remember that when a number n is multiplied by itself, we write and read it “n squared.” The result is called the square of n. For example,
Similarly, 121 is the square of 11, because is 121.
Complete the following table to show the squares of the counting numbers 1 through 15.
The numbers in the second row are called perfect square numbers. It will be helpful to learn to recognize the perfect square numbers.
The squares of the counting numbers are positive numbers. What about the squares of negative numbers? We know that when the signs of two numbers are the same, their product is positive. So the square of any negative number is also positive.
Did you notice that these squares are the same as the squares of the positive numbers?
Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because we say 100 is the square of 10. We also say that 10 is a square root of 100. A number whose square is is called a square root of m.
Notice also, so is also a square root of 100. Therefore, both 10 and are square roots of 100.
So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, denotes the positive square root. The positive square root is called the principal square root. When we use the radical sign that always means we want the principal square root.
We also use the radical sign for the square root of zero. Because Notice that zero has only one square root.
Since 10 is the principal square root of 100, we write You may want to complete the following table to help you recognize square roots.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, We read as “the opposite of the square root of 10.”
### Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
We have already described numbers as , , and . What is the difference between these types of numbers?
What type of numbers would we get if we started with all the integers and then included all the fractions? The numbers we would have form the set of rational numbers. A rational number is a number that can be written as a ratio of two integers.
All signed fractions, such as are rational numbers. Each numerator and each denominator is an integer.
Are integers rational numbers? To decide if an integer is a rational number, we try to write it as a ratio of two integers. Each integer can be written as a ratio of integers in many ways. For example, 3 is equivalent to
An easy way to write an integer as a ratio of integers is to write it as a fraction with denominator one.
Since any integer can be written as the ratio of two integers, all integers are rational numbers! Remember that the counting numbers and the whole numbers are also integers, and so they, too, are rational.
What about decimals? Are they rational? Let’s look at a few to see if we can write each of them as the ratio of two integers.
We’ve already seen that integers are rational numbers. The integer could be written as the decimal So, clearly, some decimals are rational.
Think about the decimal 7.3. Can we write it as a ratio of two integers? Because 7.3 means we can write it as an improper fraction, So 7.3 is the ratio of the integers 73 and 10. It is a rational number.
In general, any decimal that ends after a number of digits (such as 7.3 or is a rational number. We can use the reciprocal (or multiplicative inverse) of the place value of the last digit as the denominator when writing the decimal as a fraction.
Let’s look at the decimal form of the numbers we know are rational.
We have seen that every integer is a rational number, since for any integer, a. We can also change any integer to a decimal by adding a decimal point and a zero.
We have also seen that every fraction is a rational number. Look at the decimal form of the fractions we considered above.
What do these examples tell us?
Every rational number can be written both as a ratio of integers, where p and q are integers and and as a decimal that either stops or repeats.
Here are the numbers we looked at above expressed as a ratio of integers and as a decimal:
Are there any decimals that do not stop or repeat? Yes!
The number (the Greek letter pi, pronounced “pie”), which is very important in describing circles, has a decimal form that does not stop or repeat.
We can even create a decimal pattern that does not stop or repeat, such as
Numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call these numbers irrational.
Let’s summarize a method we can use to determine whether a number is rational or irrational.
We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. The irrational numbers are numbers whose decimal form does not stop and does not repeat. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.
All the numbers we use in elementary algebra are real numbers. illustrates how the number sets we’ve discussed in this section fit together.
Can we simplify Is there a number whose square is
None of the numbers that we have dealt with so far has a square that is Why? Any positive number squared is positive. Any negative number squared is positive. So we say there is no real number equal to
The square root of a negative number is not a real number.
### Locate Fractions on the Number Line
The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.
Let’s start with fractions and locate on the number line.
We’ll start with the whole numbers and because they are the easiest to plot. See .
The proper fractions listed are We know the proper fraction has value less than one and so would be located between The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts We plot See .
Similarly, is between 0 and After dividing the unit into 5 equal parts we plot See .
Finally, look at the improper fractions These are fractions in which the numerator is greater than the denominator. Locating these points may be easier if you change each of them to a mixed number. See .
shows the number line with all the points plotted.
In , we’ll use the inequality symbols to order fractions. In previous chapters we used the number line to order numbers.
1. a < b “a is less than b” when a is to the left of b on the number line
2. a > b “a is greater than b” when a is to the right of b on the number line
As we move from left to right on a number line, the values increase.
### Locate Decimals on the Number Line
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.
Which is larger, 0.04 or 0.40? If you think of this as money, you know that $0.40 (forty cents) is greater than $0.04 (four cents). So,
Again, we can use the number line to order numbers.
1. a < b “a is less than b” when a is to the left of b on the number line
2. a > b “a is greater than b” when a is to the right of b on the number line
Where are 0.04 and 0.40 located on the number line? See .
We see that 0.40 is to the right of 0.04 on the number line. This is another way to demonstrate that 0.40 > 0.04.
How does 0.31 compare to 0.308? This doesn’t translate into money to make it easy to compare. But if we convert 0.31 and 0.308 into fractions, we can tell which is larger.
Because 310 > 308, we know that Therefore, 0.31 > 0.308.
Notice what we did in converting 0.31 to a fraction—we started with the fraction and ended with the equivalent fraction Converting back to a decimal gives 0.310. So 0.31 is equivalent to 0.310. Writing zeros at the end of a decimal does not change its value!
We say 0.31 and 0.310 are equivalent decimals.
We use equivalent decimals when we order decimals.
The steps we take to order decimals are summarized here.
When we order negative decimals, it is important to remember how to order negative integers. Recall that larger numbers are to the right on the number line. For example, because lies to the right of on the number line, we know that Similarly, smaller numbers lie to the left on the number line. For example, because lies to the left of on the number line, we know that See .
If we zoomed in on the interval between 0 and as shown in , we would see in the same way that
### Key Concepts
1. Square Root Notation
is read ‘the square root of m.’ If then for
2. Order Decimals
### Practice Makes Perfect
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers
In the following exercises, identify whether each number is rational or irrational.
In the following exercises, identify whether each number is a real number or not a real number.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
In the following exercises, order each of the pairs of numbers, using < or >.
Locate Decimals on the Number Line In the following exercises, locate the number on the number line.
In the following exercises, order each pair of numbers, using < or >.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ On a scale of how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Foundations
## Properties of Real Numbers
### Use the Commutative and Associative Properties
Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?
The results are the same.
As we can see, the order in which we add does not matter!
What about multiplying
Again, the results are the same!
The order in which we multiply does not matter!
These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.
The commutative property has to do with order. If you change the order of the numbers when adding or multiplying, the result is the same.
What about subtraction? Does order matter when we subtract numbers? Does give the same result as
The results are not the same.
Since changing the order of the subtraction did not give the same result, we know that subtraction is not commutative.
Let’s see what happens when we divide two numbers. Is division commutative?
The results are not the same.
Since changing the order of the division did not give the same result, division is not commutative. The commutative properties only apply to addition and multiplication!
1. Addition and multiplication are commutative.
2. Subtraction and Division are not commutative.
If you were asked to simplify this expression, how would you do it and what would your answer be?
Some people would think and then Others might start with and then
Either way gives the same result. Remember, we use parentheses as grouping symbols to indicate which operation should be done first.
When adding three numbers, changing the grouping of the numbers gives the same result.
This is true for multiplication, too.
When multiplying three numbers, changing the grouping of the numbers gives the same result.
You probably know this, but the terminology may be new to you. These examples illustrate the associative property.
Let’s think again about multiplying We got the same result both ways, but which way was easier? Multiplying and first, as shown above on the right side, eliminates the fraction in the first step. Using the associative property can make the math easier!
The associative property has to do with grouping. If we change how the numbers are grouped, the result will be the same. Notice it is the same three numbers in the same order—the only difference is the grouping.
We saw that subtraction and division were not commutative. They are not associative either.
When simplifying an expression, it is always a good idea to plan what the steps will be. In order to combine like terms in the next example, we will use the commutative property of addition to write the like terms together.
When we have to simplify algebraic expressions, we can often make the work easier by applying the commutative or associative property first, instead of automatically following the order of operations. When adding or subtracting fractions, combine those with a common denominator first.
### Use the Identity and Inverse Properties of Addition and Multiplication
What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity.
For example,
These examples illustrate the Identity Property of Addition that states that for any real number and
What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity.
For example,
These examples illustrate the Identity Property of Multiplication that states that for any real number and
We summarize the Identity Properties below.
Notice that in each case, the missing number was the opposite of the number!
We call the additive inverse of a. The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number Remember, a number and its opposite add to zero.
What number multiplied by gives the multiplicative identity, 1? In other words, times what results in 1?
What number multiplied by 2 gives the multiplicative identity, 1? In other words 2 times what results in 1?
Notice that in each case, the missing number was the reciprocal of the number!
We call the multiplicative inverse of a. The reciprocal of A number and its reciprocal multiply to one, which is the multiplicative identity. This leads to the Inverse Property of Multiplication that states that for any real number
We’ll formally state the inverse properties here:
### Use the Properties of Zero
The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.
What about division involving zero? What is Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,
We can check division with the related multiplication fact.
So we know because
Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: means Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.
We conclude that there is no answer to and so we say that division by 0 is undefined.
We summarize the properties of zero below.
We will now practice using the properties of identities, inverses, and zero to simplify expressions.
Now we will see how recognizing reciprocals is helpful. Before multiplying left to right, look for reciprocals—their product is 1.
### Simplify Expressions Using the Distributive Property
Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?
You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.
Back to our friends at the movies, we could find the total amount of money they need like this:
In algebra, we use the distributive property to remove parentheses as we simplify expressions.
For example, if we are asked to simplify the expression the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in .
Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in would look like this:
Using the distributive property as shown in will be very useful when we solve money applications in later chapters.
When we distribute a negative number, we need to be extra careful to get the signs correct!
will show how to use the distributive property to find the opposite of an expression.
There will be times when we’ll need to use the distributive property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.
All the properties of real numbers we have used in this chapter are summarized in .
### Key Concepts
1. Commutative Property of
2. Associative Property of
3. Distributive Property: If are real numbers, then
4. Identity Property
5. Inverse Property
6. Properties of Zero
### Practice Makes Perfect
Use the Commutative and Associative Properties
In the following exercises, use the associative property to simplify.
In the following exercises, simplify.
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
In the following exercises, find the multiplicative inverse of each number.
Use the Properties of Zero
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the distributive property.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Foundations
## Systems of Measurement
### Make Unit Conversions in the U.S. System
There are two systems of measurement commonly used around the world. Most countries use the metric system. The U.S. uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart, and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, and hours.
The equivalencies of measurements are shown in . The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement, such as feet and yards, minutes and seconds, quarts and gallons, etc. We will use the identity property of multiplication to do these conversions. We’ll restate the identity property of multiplication here for easy reference.
To use the identity property of multiplication, we write 1 in a form that will help us convert the units. For example, suppose we want to change inches to feet. We know that 1 foot is equal to 12 inches, so we will write 1 as the fraction When we multiply by this fraction we do not change the value, but just change the units.
But also equals 1. How do we decide whether to multiply by or We choose the fraction that will make the units we want to convert from divide out. Treat the unit words like factors and “divide out” common units like we do common factors. If we want to convert inches to feet, which multiplication will eliminate the inches?
The inches divide out and leave only feet. The second form does not have any units that will divide out and so will not help us.
When we use the identity property of multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
Sometimes, to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.
### Use Mixed Units of Measurement in the U.S. System
We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.
Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units!
### Make Unit Conversions in the Metric System
In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix kilo means thousand. One centimeter is of a meter, just like one cent is of one dollar.
The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.
To make conversions in the metric system, we will use the same technique we did in the US system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a 5K or 10K race? The length of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.
As you become familiar with the metric system you may see a pattern. Since the system is based on multiples of ten, the calculations involve multiplying by multiples of ten. We have learned how to simplify these calculations by just moving the decimal.
To multiply by 10, 100, or 1,000, we move the decimal to the right one, two, or three places, respectively. To multiply by 0.1, 0.01, or 0.001, we move the decimal to the left one, two, or three places, respectively.
We can apply this pattern when we make measurement conversions in the metric system. In , we changed 3,200 grams to kilograms by multiplying by (or 0.001). This is the same as moving the decimal three places to the left.
### Use Mixed Units of Measurement in the Metric System
Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.
### Convert Between the U.S. and the Metric Systems of Measurement
Many measurements in the United States are made in metric units. Our soda may come in 2-liter bottles, our calcium may come in 500-mg capsules, and we may run a 5K race. To work easily in both systems, we need to be able to convert between the two systems.
shows some of the most common conversions.
shows how inches and centimeters are related on a ruler.
shows the ounce and milliliter markings on a measuring cup.
shows how pounds and kilograms marked on a bathroom scale.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.
### Convert between Fahrenheit and Celsius Temperatures
Have you ever been in a foreign country and heard the weather forecast? If the forecast is for what does that mean?
The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written The metric system uses degrees Celsius, written shows the relationship between the two systems.
### Key Concepts
1. Metric System of Measurement
2. Temperature Conversion
### Section Exercises
### Practice Makes Perfect
Make Unit Conversions in the U.S. System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the U.S. System
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exercises, solve.
Convert Between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next Chapter? Why or why not?
### Chapter Review Exercises
### Introduction to Whole Numbers
Use Place Value with Whole Number
In the following exercises find the place value of each digit.
In the following exercises, name each number.
In the following exercises, write each number as a whole number using digits.
In the following exercises, round to the indicated place value.
In the following exercises, round each number to the nearest ⓐ hundred ⓑ thousand ⓒ ten thousand.
Identify Multiples and Factors
In the following exercises, use the divisibility tests to determine whether each number is divisible by 2, by 3, by 5, by 6, and by 10.
Find Prime Factorizations and Least Common Multiples
In the following exercises, find the prime factorization.
In the following exercises, find the least common multiple of the following numbers using the multiples method.
In the following exercises, find the least common multiple of the following numbers using the prime factors method.
### Use the Language of Algebra
Use Variables and Algebraic Symbols
In the following exercises, translate the following from algebra to English.
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
In the following exercises, simplify
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
In the following exercises, identify the terms in each expression.
In the following exercises, simplify the following expressions by combining like terms.
Translate an English Phrase to an Algebraic Expression
In the following exercises, translate the following phrases into algebraic expressions.
### Add and Subtract Integers
Use Negatives and Opposites of Integers
In the following exercises, order each of the following pairs of numbers, using < or >.
In the following exercises,, find the opposite of each number.
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions with Absolute Value
In the following exercises,, simplify.
In the following exercises, fill in <, >, or = for each of the following pairs of numbers.
In the following exercises, simplify.
In the following exercises, evaluate.
Add Integers
In the following exercises, simplify each expression.
Subtract Integers
In the following exercises, simplify.
In the following exercises, simplify each expression.
Multiply Integers
In the following exercises, multiply.
Divide Integers
In the following exercises, divide.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate to an algebraic expression and simplify if possible.
Use Integers in Applications
In the following exercises, solve.
### Visualize Fractions
Find Equivalent Fractions
In the following exercises, find three fractions equivalent to the given fraction. Show your work, using figures or algebra.
Simplify Fractions
In the following exercises, simplify.
Multiply Fractions
In the following exercises, multiply.
Divide Fractions
In the following exercises, divide.
In the following exercises, simplify.
Simplify Expressions Written with a Fraction Bar
In the following exercises, simplify.
Translate Phrases to Expressions with Fractions
In the following exercises, translate each English phrase into an algebraic expression.
### Add and Subtract Fractions
Add and Subtract Fractions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Add or Subtract Fractions with Different Denominators
In the following exercises, add or subtract.
Use the Order of Operations to Simplify Complex Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
### Decimals
Name and Write Decimals
In the following exercises, write as a decimal.
In the following exercises, name each decimal.
Round Decimals
In the following exercises, round each number to the nearest ⓐ hundredth ⓑ tenth ⓒ whole number.
Add and Subtract Decimals
In the following exercises, add or subtract.
Multiply and Divide Decimals
In the following exercises, multiply.
In the following exercises, divide.
Convert Decimals, Fractions, and Percents
In the following exercises, write each decimal as a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each percent to a decimal.
In the following exercises, convert each decimal to a percent.
### The Real Numbers
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write as the ratio of two integers.
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers.
In the following exercises, identify whether each number is rational or irrational.
In the following exercises, identify whether each number is a real number or not a real number.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
In the following exercises, order each of the following pairs of numbers, using < or >.
Locate Decimals on the Number Line
In the following exercises, locate on the number line.
In the following exercises, order each of the following pairs of numbers, using < or >.
### Properties of Real Numbers
Use the Commutative and Associative Properties
In the following exercises, use the Associative Property to simplify.
In the following exercises, simplify.
Use the Identity and Inverse Properties of Addition and Multiplication
In the following exercises, find the additive inverse of each number.
In the following exercises, find the multiplicative inverse of each number.
Use the Properties of Zero
In the following exercises, simplify.
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.
### Systems of Measurement
1.1 Define U.S. Units of Measurement and Convert from One Unit to Another
In the following exercises, convert the units. Round to the nearest tenth.
Use Mixed Units of Measurement in the U.S. System.
In the following exercises, solve.
Make Unit Conversions in the Metric System
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System
In the following exerices, solve.
Convert between the U.S. and the Metric Systems of Measurement
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius Temperatures
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.
### Chapter Practice Test
In the following exercises, evaluate.
In the following exercises, simplify each expression. |
# Solving Linear Equations and Inequalities
## Introduction
If we carefully placed more rocks of equal weight on both sides of this formation, it would still balance. Similarly, the expressions in an equation remain balanced when we add the same quantity to both sides of the equation. In this chapter, we will solve equations, remembering that what we do to one side of the equation, we must also do to the other side. |
# Solving Linear Equations and Inequalities
## Solve Equations Using the Subtraction and Addition Properties of Equality
### Verify a Solution of an Equation
Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!
### Solve Equations Using the Subtraction and Addition Properties of Equality
We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in . Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?
What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?
Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See for an illustration of this process.
What algebraic equation would match this situation? In each side of the workspace represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope .
Let’s write algebraically the steps we took to discover how many counters were in the envelope:
Check:
Five in the envelope plus three more does equal eight!
Our model has given us an idea of what we need to do to solve one kind of equation. The goal is to isolate the variable by itself on one side of the equation. To solve equations such as these mathematically, we use the Subtraction Property of Equality.
Let’s see how to use this property to solve an equation. Remember, the goal is to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.
What happens when an equation has a number subtracted from the variable, as in the equation ? We use another property of equations to solve equations where a number is subtracted from the variable. We want to isolate the variable, so to ‘undo’ the subtraction we will add the number to both sides. We use the Addition Property of Equality.
In , 37 was added to the y and so we subtracted 37 to ‘undo’ the addition. In , we will need to ‘undo’ subtraction by using the Addition Property of Equality.
The next example will be an equation with decimals.
### Solve Equations That Require Simplification
In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.
You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.
### Translate to an Equation and Solve
To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign. shows us some of the words that are commonly used.
The steps we use to translate a sentence into an equation are listed below.
### Translate and Solve Applications
Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.
We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.
### Key Concepts
1. To Determine Whether a Number is a Solution to an Equation
2. Addition Property of Equality
3. Subtraction Property of Equality
4. To Translate a Sentence to an Equation
5. To Solve an Application
### Practice Makes Perfect
Verify a Solution of an Equation
In the following exercises, determine whether the given value is a solution to the equation.
Solve Equations using the Subtraction and Addition Properties of Equality
In the following exercises, solve each equation using the Subtraction and Addition Properties of Equality.
Solve Equations that Require Simplification
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve it.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Solving Linear Equations and Inequalities
## Solve Equations using the Division and Multiplication Properties of Equality
### Solve Equations Using the Division and Multiplication Properties of Equality
You may have noticed that all of the equations we have solved so far have been of the form or . We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.
Let’s look at our puzzle again with the envelopes and counters in .
In the illustration there are two identical envelopes that contain the same number of counters. Remember, the left side of the workspace must equal the right side, but the counters on the left side are “hidden” in the envelopes. So how many counters are in each envelope?
How do we determine the number? We have to separate the counters on the right side into two groups of the same size to correspond with the two envelopes on the left side. The 6 counters divided into 2 equal groups gives 3 counters in each group (since ).
What equation models the situation shown in ? There are two envelopes, and each contains counters. Together, the two envelopes must contain a total of 6 counters.
We found that each envelope contains 3 counters. Does this check? We know , so it works! Three counters in each of two envelopes does equal six!
This example leads to the Division Property of Equality.
The goal in solving an equation is to ‘undo’ the operation on the variable. In the next example, the variable is multiplied by 5, so we will divide both sides by 5 to ‘undo’ the multiplication.
Consider the equation . We want to know what number divided by 4 gives 3. So to “undo” the division, we will need to multiply by 4. The Multiplication Property of Equality will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.
In the next example, all the variable terms are on the right side of the equation. As always, our goal in solving the equation is to isolate the variable.
### Solve Equations That Require Simplification
Many equations start out more complicated than the ones we have been working with.
With these more complicated equations the first step is to simplify both sides of the equation as much as possible. This usually involves combining like terms or using the distributive property.
Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.
### Translate to an Equation and Solve
In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.
### Translate and Solve Applications
To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.
### Key Concepts
1. The Division Property of Equality—For any numbers a, b, and c, and , if , then .
When you divide both sides of an equation by any non-zero number, you still have equality.
2. The Multiplication Property of Equality—For any numbers a, b, and c, if , then .
If you multiply both sides of an equation by the same number, you still have equality.
### Practice Makes Perfect
Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the Division and Multiplication Properties of Equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Mixed Practice
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Solving Linear Equations and Inequalities
## Solve Equations with Variables and Constants on Both Sides
### Solve Equations with Constants on Both Sides
In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.
Our strategy will involve choosing one side of the equation to be the “variable side”, and the other side of the equation to be the “constant side.” Then, we will use the Subtraction and Addition Properties of Equality to get all the variable terms together on one side of the equation and the constant terms together on the other side.
By doing this, we will transform the equation that began with variables and constants on both sides into the form We already know how to solve equations of this form by using the Division or Multiplication Properties of Equality.
### Solve Equations with Variables on Both Sides
What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.
### Solve Equations with Variables and Constants on Both Sides
The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.
We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this chapter.
In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.
In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!
To solve an equation with fractions, we just follow the steps of our strategy to get the solution!
We will use the same strategy to find the solution for an equation with decimals.
### Key Concepts
1. Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation
### Practice Makes Perfect
Solve Equations with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve Equations with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve Equations with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Solving Linear Equations and Inequalities
## Use a General Strategy to Solve Linear Equations
### Solve Equations Using the General Strategy
Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.
Beginning by simplifying each side of the equation makes the remaining steps easier.
### Classify Equations
Consider the equation we solved at the start of the last section, . The solution we found was . This means the equation is true when we replace the variable, x, with the value . We showed this when we checked the solution and evaluated for .
If we evaluate for a different value of x, the left side will not be .
The equation is true when we replace the variable, x, with the value , but not true when we replace x with any other value. Whether or not the equation is true depends on the value of the variable. Equations like this are called conditional equations.
All the equations we have solved so far are conditional equations.
Now let’s consider the equation . Do you recognize that the left side and the right side are equivalent? Let’s see what happens when we solve for y.
But is true.
This means that the equation is true for any value of y. We say the solution to the equation is all of the real numbers. An equation that is true for any value of the variable like this is called an identity.
What happens when we solve the equation ?
But .
Solving the equation led to the false statement . The equation will not be true for any value of z. It has no solution. An equation that has no solution, or that is false for all values of the variable, is called a contradiction.
### Key Concepts
1. General Strategy for Solving Linear Equations
### Practice Makes Perfect
Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Solving Linear Equations and Inequalities
## Solve Equations with Fractions or Decimals
### Solve Equations with Fraction Coefficients
Let’s use the general strategy for solving linear equations introduced earlier to solve the equation, .
This method worked fine, but many students do not feel very confident when they see all those fractions. So, we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.
We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but without fractions. This process is called “clearing” the equation of fractions.
Let’s solve a similar equation, but this time use the method that eliminates the fractions.
Notice in , once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.
In the next example, we again have variables on both sides of the equation.
In the next example, we start by using the Distributive Property. This step clears the fractions right away.
In the next example, even after distributing, we still have fractions to clear.
### Solve Equations with Decimal Coefficients
Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, and . So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.
The next example uses an equation that is typical of the money applications in the next chapter. Notice that we distribute the decimal before we clear all the decimals.
### Key Concepts
1. Strategy to Solve an Equation with Fraction Coefficients
### Practice Makes Perfect
Solve Equations with Fraction Coefficients
In the following exercises, solve each equation with fraction coefficients.
Solve Equations with Decimal Coefficients
In the following exercises, solve each equation with decimal coefficients.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Solving Linear Equations and Inequalities
## Solve a Formula for a Specific Variable
### Use the Distance, Rate, and Time Formula
One formula you will use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant rate. Rate is an equivalent word for “speed.” The basic idea of rate may already familiar to you. Do you know what distance you travel if you drive at a steady rate of 60 miles per hour for 2 hours? (This might happen if you use your car’s cruise control while driving on the highway.) If you said 120 miles, you already know how to use this formula!
We will use the Strategy for Solving Applications that we used earlier in this chapter. When our problem requires a formula, we change Step 4. In place of writing a sentence, we write the appropriate formula. We write the revised steps here for reference.
You may want to create a mini-chart to summarize the information in the problem. See the chart in this first example.
### Solve a Formula for a Specific Variable
You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.
In and , we used the formula . This formula gives the value of , distance, when you substitute in the values of , the rate and time. But in , we had to find the value of . We substituted in values of and then used algebra to solve for . If you had to do this often, you might wonder why there is not a formula that gives the value of when you substitute in the values of . We can make a formula like this by solving the formula for .
To solve a formula for a specific variable means to isolate that variable on one side of the equals sign with a coefficient of 1. All other variables and constants are on the other side of the equals sign. To see how to solve a formula for a specific variable, we will start with the distance, rate and time formula.
The formula is used to calculate simple interest, I, for a principal, P, invested at rate, r, for t years.
Later in this class, and in future algebra classes, you’ll encounter equations that relate two variables, usually x and y. You might be given an equation that is solved for y and need to solve it for x, or vice versa. In the following example, we’re given an equation with both x and y on the same side and we’ll solve it for y.
In Examples 1.60 through 1.64 we used the numbers in part ⓐ as a guide to solving in general in part ⓑ. Now we will solve a formula in general without using numbers as a guide.
### Key Concepts
1. To Solve an Application (with a formula)
2. Distance, Rate and Time
For an object moving at a uniform (constant) rate, the distance traveled, the elapsed time, and the rate are related by the formula: where d = distance, r = rate, t = time.
3. To solve a formula for a specific variable means to get that variable by itself with a coefficient of 1 on one side of the equation and all other variables and constants on the other side.
### Practice Makes Perfect
Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, use the formula .
In the following exercises, use the formula .
In the following exercises, use the formula I = Prt.
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Solving Linear Equations and Inequalities
## Solve Linear Inequalities
### Graph Inequalities on the Number Line
Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
What about the solution of an inequality? What number would make the inequality true? Are you thinking, ‘x could be 4’? That’s correct, but x could be 5 too, or 20, or even 3.001. Any number greater than 3 is a solution to the inequality .
We show the solutions to the inequality on the number line by shading in all the numbers to the right of 3, to show that all numbers greater than 3 are solutions. Because the number 3 itself is not a solution, we put an open parenthesis at 3. The graph of is shown in . Please note that the following convention is used: light blue arrows point in the positive direction and dark blue arrows point in the negative direction.
The graph of the inequality is very much like the graph of , but now we need to show that 3 is a solution, too. We do that by putting a bracket at , as shown in .
Notice that the open parentheses symbol, (, shows that the endpoint of the inequality is not included. The open bracket symbol, [, shows that the endpoint is included.
We can also represent inequalities using interval notation. As we saw above, the inequality means all numbers greater than 3. There is no upper end to the solution to this inequality. In interval notation, we express as The symbol is read as ‘infinity’. It is not an actual number. shows both the number line and the interval notation.
The inequality means all numbers less than or equal to 1. There is no lower end to those numbers. We write in interval notation as . The symbol is read as ‘negative infinity’. shows both the number line and interval notation.
Did you notice how the parenthesis or bracket in the interval notation matches the symbol at the endpoint of the arrow? These relationships are shown in .
### Solve Inequalities using the Subtraction and Addition Properties of Inequality
The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.
Similar properties hold true for inequalities.
Similarly we could show that the inequality also stays the same for addition.
This leads us to the Subtraction and Addition Properties of Inequality.
We use these properties to solve inequalities, taking the same steps we used to solve equations. Solving the inequality , the steps would look like this:
Any number greater than 4 is a solution to this inequality.
### Solve Inequalities using the Division and Multiplication Properties of Inequality
The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).
Are there similar properties for inequalities? What happens to an inequality when we divide or multiply both sides by a constant?
Consider some numerical examples.
Does the inequality stay the same when we divide or multiply by a negative number?
When we divide or multiply an inequality by a positive number, the inequality sign stays the same. When we divide or multiply an inequality by a negative number, the inequality sign reverses.
Here are the Division and Multiplication Properties of Inequality for easy reference.
When we divide or multiply an inequality by a:
1. positive number, the inequality stays the same.
2. negative number, the inequality reverses.
### Solve Inequalities That Require Simplification
Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.
Just like some equations are identities and some are contradictions, inequalities may be identities or contradictions, too. We recognize these forms when we are left with only constants as we solve the inequality. If the result is a true statement, we have an identity. If the result is a false statement, we have a contradiction.
### Translate to an Inequality and Solve
To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.
Think about the phrase ‘at least’ – what does it mean to be ‘at least 21 years old’? It means 21 or more. The phrase ‘at least’ is the same as ‘greater than or equal to’.
shows some common phrases that indicate inequalities.
### Key Concepts
1. Subtraction Property of Inequality
For any numbers a, b, and c,
if then and
if then
2. Addition Property of Inequality
For any numbers a, b, and c,
if then and
if then
3. Division and Multiplication Properties of Inequality
For any numbers a, b, and c,
if and , then and .
if and , then and .
if and , then and .
if and , then and .
4. When we divide or multiply an inequality by a:
### Section Exercises
### Practice Makes Perfect
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Mixed practice
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate to an Inequality and Solve
In the following exercises, translate and solve .Then write the solution in interval notation and graph on the number line.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
### Chapter 2 Review Exercises
### Solve Equations using the Subtraction and Addition Properties of Equality
Verify a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
Solve Equations using the Subtraction and Addition Properties of Equality
In the following exercises, solve each equation using the Subtraction Property of Equality.
In the following exercises, solve each equation using the Addition Property of Equality.
In the following exercises, solve each equation.
Solve Equations That Require Simplification
In the following exercises, solve each equation.
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Translate and Solve Applications
In the following exercises, translate into an algebraic equation and solve.
### Solve Equations using the Division and Multiplication Properties of Equality
Solve Equations Using the Division and Multiplication Properties of Equality
In the following exercises, solve each equation using the division and multiplication properties of equality and check the solution.
Solve Equations That Require Simplification
In the following exercises, solve each equation requiring simplification.
Translate to an Equation and Solve
In the following exercises, translate to an equation and then solve.
Translate and Solve Applications
In the following exercises, translate into an equation and solve.
### Solve Equations with Variables and Constants on Both Sides
Solve an Equation with Constants on Both Sides
In the following exercises, solve the following equations with constants on both sides.
Solve an Equation with Variables on Both Sides
In the following exercises, solve the following equations with variables on both sides.
Solve an Equation with Variables and Constants on Both Sides
In the following exercises, solve the following equations with variables and constants on both sides.
### Use a General Strategy for Solving Linear Equations
Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, solve each linear equation.
Classify Equations
In the following exercises, classify each equation as a conditional equation, an identity, or a contradiction and then state the solution.
### Solve Equations with Fractions and Decimals
Solve Equations with Fraction Coefficients
In the following exercises, solve each equation with fraction coefficients.
Solve Equations with Decimal Coefficients
In the following exercises, solve each equation with decimal coefficients.
### Solve a Formula for a Specific Variable
Use the Distance, Rate, and Time Formula
In the following exercises, solve.
Solve a Formula for a Specific Variable
In the following exercises, solve.
### Solve Linear Inequalities
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line.
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Inequalities using the Subtraction and Addition Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities using the Division and Multiplication Properties of Inequality
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Inequalities That Require Simplification
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate to an Inequality and Solve
In the following exercises, translate and solve. Then write the solution in interval notation and graph on the number line.
### Everyday Math
### Chapter 2 Practice Test
In the following exercises, solve each equation.
In the following exercises, graph on the number line and write in interval notation.
In the following exercises,, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
In the following exercises, translate to an equation or inequality and solve. |
# Math Models
## Introduction
Mathematical formulas model phenomena in every facet of our lives. They are used to explain events and predict outcomes in fields such as transportation, business, economics, medicine, chemistry, engineering, and many more. In this chapter, we will apply our skills in solving equations to solve problems in a variety of situations. |
# Math Models
## Use a Problem-Solving Strategy
### Approach Word Problems with a Positive Attitude
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
### Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
Let’s try this approach with another example.
### Solve Number Problems
Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as n, the next consecutive integer is The one after that is one more than so it is which is
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
Notice each integer is 2 more than the number preceding it. If we call the first one n, then the next one is The next one would be or
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.
Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.
### Key Concepts
1. Problem-Solving Strategy
2. Consecutive Integers
Consecutive integers are integers that immediately follow each other.
Consecutive even integers are even integers that immediately follow one another.
Consecutive odd integers are odd integers that immediately follow one another.
### Practice Makes Perfect
Use the Approach Word Problems with a Positive Attitude
In the following exercises, prepare the lists described.
Use a Problem-Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Problems
In the following exercises, solve each number word problem.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Math Models
## Solve Percent Applications
### Translate and Solve Basic Percent Equations
We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.
We can use any letter you like as a variable, but it is a good idea to choose a letter that will remind us of what you are looking for. We must be sure to change the given percent to a decimal when we put it in the equation.
We must be very careful when we translate the words in the next example. The unknown quantity will not be isolated at first, like it was in . We will again use direct translation to write the equation.
In the next example, we are looking for the percent.
### Solve Applications of Percent
Many applications of percent—such as tips, sales tax, discounts, and interest—occur in our daily lives. To solve these applications we’ll translate to a basic percent equation, just like those we solved in previous examples. Once we translate the sentence into a percent equation, we know how to solve it.
We will restate the problem solving strategy we used earlier for easy reference.
Now that we have the strategy to refer to, and have practiced solving basic percent equations, we are ready to solve percent applications. Be sure to ask yourself if your final answer makes sense—since many of the applications will involve everyday situations, you can rely on your own experience.
### Find Percent Increase and Percent Decrease
People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.
To find the percent increase, first we find the amount of increase, the difference of the new amount and the original amount. Then we find what percent the amount of increase is of the original amount.
Finding the percent decrease is very similar to finding the percent increase, but now the amount of decrease is the difference of the original amount and the new amount. Then we find what percent the amount of decrease is of the original amount.
### Solve Simple Interest Applications
Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, P, and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; called the rate of interest, r. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is in the account.
To find the interest we use the simple interest formula,
Interest may also be calculated another way, called compound interest. This type of interest will be covered in later math classes.
The formula we use to calculate simple interest is To use the formula, we substitute in the values the problem gives us for the variables, and then solve for the unknown variable. It may be helpful to organize the information in a chart.
There may be times when we know the amount of interest earned on a given principal over a certain length of time, but we don’t know the rate. To find the rate, we use the simple interest formula, substitute in the given values for the principal and time, and then solve for the rate.
Notice that in this example, Loren’s brother paid Loren interest, just like a bank would have paid interest if Loren invested his money there.
### Solve Applications with Discount or Mark-up
Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.
We summarize the discount model in the box below.
There may be times when we know the original price and the sale price, and we want to know the discount rate. To find the discount rate, first we will find the amount of discount and then use it to compute the rate as a percent of the original price. will show this case.
Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the original cost. The retailer then adds a mark-up to the original cost to get the list price, the price he sells the item for. The mark-up is usually calculated as a percent of the original cost. To determine the amount of mark-up, multiply the mark-up rate by the original cost.
We summarize the mark-up model in the box below.
### Key Concepts
1. Percent Increase To find the percent increase:
2. Percent Decrease To find the percent decrease:
3. Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is
4. Discount
5. Mark-up
### Practice Makes Perfect
Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Percent Applications
In the following exercises, solve.
Find Percent Increase and Percent Decrease
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
Solve Applications with Discount or Mark-up
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals? |
# Math Models
## Solve Mixture Applications
### Solve Coin Word Problems
In mixture problems, we will have two or more items with different values to combine together. The mixture model is used by grocers and bartenders to make sure they set fair prices for the products they sell. Many other professionals, like chemists, investment bankers, and landscapers also use the mixture model.
We will start by looking at an application everyone is familiar with—money!
Imagine that we take a handful of coins from a pocket or purse and place them on a desk. How would we determine the value of that pile of coins? If we can form a step-by-step plan for finding the total value of the coins, it will help us as we begin solving coin word problems.
So what would we do? To get some order to the mess of coins, we could separate the coins into piles according to their value. Quarters would go with quarters, dimes with dimes, nickels with nickels, and so on. To get the total value of all the coins, we would add the total value of each pile.
How would we determine the value of each pile? Think about the dime pile—how much is it worth? If we count the number of dimes, we’ll know how many we have—the number of dimes.
But this does not tell us the value of all the dimes. Say we counted 17 dimes, how much are they worth? Each dime is worth $0.10—that is the value of one dime. To find the total value of the pile of 17 dimes, multiply 17 by $0.10 to get $1.70. This is the total value of all 17 dimes. This method leads to the following model.
The number of dimes times the value of each dime equals the total value of the dimes.
We could continue this process for each type of coin, and then we would know the total value of each type of coin. To get the total value of all the coins, add the total value of each type of coin.
Let’s look at a specific case. Suppose there are 14 quarters, 17 dimes, 21 nickels, and 39 pennies.
The total value of all the coins is $6.64.
Notice how the chart helps organize all the information! Let’s see how we use this method to solve a coin word problem.
In the next example, we’ll show only the completed table—remember the steps we take to fill in the table.
### Solve Ticket and Stamp Word Problems
Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.
We have learned how to find the total number of tickets when the number of one type of ticket is based on the number of the other type. Next, we’ll look at an example where we know the total number of tickets and have to figure out how the two types of tickets relate.
Suppose Bianca sold a total of 100 tickets. Each ticket was either an adult ticket or a child ticket. If she sold 20 child tickets, how many adult tickets did she sell?
1. Did you say ‘80’? How did you figure that out? Did you subtract 20 from 100?
If she sold 45 child tickets, how many adult tickets did she sell?
1. Did you say ‘55’? How did you find it? By subtracting 45 from 100?
What if she sold 75 child tickets? How many adult tickets did she sell?
1. The number of adult tickets must be
Now, suppose Bianca sold x child tickets. Then how many adult tickets did she sell? To find out, we would follow the same logic we used above. In each case, we subtracted the number of child tickets from 100 to get the number of adult tickets. We now do the same with x.
We have summarized this below.
We can apply these techniques to other examples
Now, we’ll do one where we fill in the table all at once.
### Solve Mixture Word Problems
Now we’ll solve some more general applications of the mixture model. Grocers and bartenders use the mixture model to set a fair price for a product made from mixing two or more ingredients. Financial planners use the mixture model when they invest money in a variety of accounts and want to find the overall interest rate. Landscape designers use the mixture model when they have an assortment of plants and a fixed budget, and event coordinators do the same when choosing appetizers and entrees for a banquet.
Our first mixture word problem will be making trail mix from raisins and nuts.
We can also use the mixture model to solve investment problems using simple interest. We have used the simple interest formula, where represented the number of years. When we just need to find the interest for one year, so then
### Key Concepts
1. Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model.
where number is the number of coins and value is the value of each coin; total value is the total value of all the coins
2. Problem-Solving Strategy—Coin Word Problems
### Practice Makes Perfect
Solve Coin Word Problems
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Solve Mixture Word Problems
In the following exercises, solve each mixture word problem.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Math Models
## Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
### Solve Applications Using Properties of Triangles
In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.
We will start geometry applications by looking at the properties of triangles. Let’s review some basic facts about triangles. Triangles have three sides and three interior angles. Usually each side is labeled with a lowercase letter to match the uppercase letter of the opposite vertex.
The plural of the word vertex is vertices. All triangles have three vertices. Triangles are named by their vertices: The triangle in is called
The three angles of a triangle are related in a special way. The sum of their measures is Note that we read as “the measure of angle A.” So in in ,
Because the perimeter of a figure is the length of its boundary, the perimeter of is the sum of the lengths of its three sides.
To find the area of a triangle, we need to know its base and height. The height is a line that connects the base to the opposite vertex and makes a angle with the base. We will draw again, and now show the height, h. See .
The triangle properties we used so far apply to all triangles. Now we will look at one specific type of triangle—a right triangle. A right triangle has one angle, which we usually mark with a small square in the corner.
In the examples we have seen so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. We will wait to draw the figure until we write expressions for all the angles we are looking for.
### Use the Pythagorean Theorem
We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem. This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.
Before we state the Pythagorean Theorem, we need to introduce some terms for the sides of a triangle. Remember that a right triangle has a angle, marked with a small square in the corner. The side of the triangle opposite the angle is called the hypotenuse and each of the other sides are called legs.
The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the lengths of the two legs equals the square of the length of the hypotenuse. In symbols we say: in any right triangle, where are the lengths of the legs and is the length of the hypotenuse.
Writing the formula in every exercise and saying it aloud as you write it, may help you remember the Pythagorean Theorem.
To solve exercises that use the Pythagorean Theorem, we will need to find square roots. We have used the notation and the definition:
If then for
For example, we found that is 5 because
Because the Pythagorean Theorem contains variables that are squared, to solve for the length of a side in a right triangle, we will have to use square roots.
### Solve Applications Using Rectangle Properties
You may already be familiar with the properties of rectangles. Rectangles have four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, L, and its adjacent side as the width, W.
The distance around this rectangle is or This is the perimeter, P, of the rectangle.
What about the area of a rectangle? Imagine a rectangular rug that is 2-feet long by 3-feet wide. Its area is 6 square feet. There are six squares in the figure.
The area is the length times the width.
The formula for the area of a rectangle is
We have solved problems where either the length or width was given, along with the perimeter or area; now we will learn how to solve problems in which the width is defined in terms of the length. We will wait to draw the figure until we write an expression for the width so that we can label one side with that expression.
### Key Concepts
1. Problem-Solving Strategy for Geometry Applications
2. Triangle Properties For
Angle measures:
Perimeter:
Area:
A right triangle has one angle.
3. The Pythagorean Theorem In any right triangle, where c is the length of the hypotenuse and a and b are the lengths of the legs.
4. Properties of Rectangles
### Practice Makes Perfect
Solving Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the leg. Round to the nearest tenth, if necessary.
In the following exercises, solve using the Pythagorean Theorem. Approximate to the nearest tenth, if necessary.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Math Models
## Solve Uniform Motion Applications
### Solve Uniform Motion Applications
When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula, which we have already seen.
In this section, we will use this formula in situations that require a little more algebra to solve than the ones we saw earlier. Generally, we will be looking at comparing two scenarios, such as two vehicles travelling at different rates or in opposite directions. When the speed of each vehicle is constant, we call applications like this uniform motion problems.
Our problem-solving strategies will still apply here, but we will add to the first step. The first step will include drawing a diagram that shows what is happening in the example. Drawing the diagram helps us understand what is happening so that we will write an appropriate equation. Then we will make a table to organize the information, like we did for the money applications.
The steps are listed here for easy reference:
In , the last example, we had two trains traveling the same distance. The diagram and the chart helped us write the equation we solved. Let’s see how this works in another case.
As you read the next example, think about the relationship of the distances traveled. Which of the previous two examples is more similar to this situation?
In the distance, rate, and time formula, time represents the actual amount of elapsed time (in hours, minutes, etc.). If a problem gives us starting and ending times as clock times, we must find the elapsed time in order to use the formula.
### Key Concepts
1. Distance, Rate, and Time
2. Problem-Solving Strategy—Distance, Rate, and Time Applications
### Practice Makes Perfect
Solve Uniform Motion Applications
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Math Models
## Solve Applications with Linear Inequalities
### Solve Applications with Linear Inequalities
Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for $20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?
The method we will use to solve applications with linear inequalities is very much like the one we used when we solved applications with equations. We will read the problem and make sure all the words are understood. Next, we will identify what we are looking for and assign a variable to represent it. We will restate the problem in one sentence to make it easy to translate into an inequality. Then, we will solve the inequality.
Sometimes an application requires the solution to be a whole number, but the algebraic solution to the inequality is not a whole number. In that case, we must round the algebraic solution to a whole number. The context of the application will determine whether we round up or down. To check applications like this, we will round our answer to a number that is easy to compute with and make sure that number makes the inequality true.
A common goal of most businesses is to make a profit. Profit is the money that remains when the expenses have been subtracted from the money earned. In the next example, we will find the number of jobs a small businessman needs to do every month in order to make a certain amount of profit.
Sometimes life gets complicated! There are many situations in which several quantities contribute to the total expense. We must make sure to account for all the individual expenses when we solve problems like this.
### Key Concepts
1. Solving inequalities
### Section Exercises
### Practice Makes Perfect
Solve Applications with Linear Inequalities
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
### Chapter 3 Review Exercises
### 3.1 Using a Problem Solving Strategy
Approach Word Problems with a Positive Attitude
In the following exercises, reflect on your approach to word problems.
Use a Problem-Solving Strategy for Word Problems
In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.
Solve Number Problems
In the following exercises, solve each number word problem.
### 3.2 Solve Percent Applications
Translate and Solve Basic Percent Equations
In the following exercises, translate and solve.
Solve Percent Applications
In the following exercises, solve.
Find Percent Increase and Percent Decrease
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
Solve Applications with Discount or Mark-up
In the following exercises, find the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.
In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)
In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.
### 3.3 Solve Mixture Applications
Solve Coin Word Problems
In the following exercises, solve each coin word problem.
Solve Ticket and Stamp Word Problems
In the following exercises, solve each ticket or stamp word problem.
Solve Mixture Word Problems
In the following exercises, solve each mixture word problem.
### 3.4 Solve Geometry Applications: Triangles, Rectangles and the Pythagorean Theorem
Solve Applications Using Triangle Properties
In the following exercises, solve using triangle properties.
Use the Pythagorean Theorem
In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.
In the following exercises, solve. Approximate to the nearest tenth, if necessary.
Solve Applications Using Rectangle Properties
In the following exercises, solve using rectangle properties.
### 3.5 Solve Uniform Motion Applications
Solve Uniform Motion Applications
In the following exercises, solve.
### 3.6 Solve Applications with Linear Inequalities
Solve Applications with Linear Inequalities
In the following exercises, solve.
### Practice Test
In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary. |
# Graphs
## Introduction
Graphs are found in all areas of our lives—from commercials showing you which cell phone carrier provides the best coverage, to bank statements and news articles, to the boardroom of major corporations. In this chapter, we will study the rectangular coordinate system, which is the basis for most consumer graphs. We will look at linear graphs, slopes of lines, equations of lines, and linear inequalities. |
# Graphs
## Use the Rectangular Coordinate System
### Plot Points on a Rectangular Coordinate System
Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the ‘coordinate plane’.
The horizontal number line is called the x-axis. The vertical number line is called the y-axis. The x-axis and the y-axis together form the rectangular coordinate system. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the of the point, and the second number is the of the point.
The phrase ‘ordered pair’ means the order is important. What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is . The point has a special name. It is called the origin.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through . Then, locate 3 on the y-axis and sketch a horizontal line through . Now, find the point where these two lines meet—that is the point with coordinates .
Notice that the vertical line through and the horizontal line through are not part of the graph. We just used them to help us locate the point .
How do the signs affect the location of the points? You may have noticed some patterns as you graphed the points in the previous example.
For the point in in Quadrant IV, what do you notice about the signs of the coordinates? What about the signs of the coordinates of points in the third quadrant? The second quadrant? The first quadrant?
Can you tell just by looking at the coordinates in which quadrant the point is located? In which quadrant is located?
What if one coordinate is zero as shown in ? Where is the point located? Where is the point located?
The point is on the y-axis and the point is on the x-axis.
In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the x-coordinate of a point on a graph, read the number on the x-axis directly above or below the point. To identify the y-coordinate of a point, read the number on the y-axis directly to the left or right of the point. Remember, when you write the ordered pair use the correct order, .
### Verify Solutions to an Equation in Two Variables
Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like . (Then, you checked the solution by substituting back into the equation.)
Here’s an example of an equation in one variable, and its one solution.
But equations can have more than one variable. Equations with two variables may be of the form . Equations of this form are called linear equations in two variables.
Notice the word line in linear. Here is an example of a linear equation in two variables, and .
The equation is also a linear equation. But it does not appear to be in the form . We can use the Addition Property of Equality and rewrite it in form.
By rewriting as , we can easily see that it is a linear equation in two variables because it is of the form . When an equation is in the form , we say it is in standard form.
Most people prefer to have , , and be integers and when writing a linear equation in standard form, although it is not strictly necessary.
Linear equations have infinitely many solutions. For every number that is substituted for there is a corresponding value. This pair of values is a solution to the linear equation and is represented by the ordered pair . When we substitute these values of and into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.
### Complete a Table of Solutions to a Linear Equation in Two Variables
In the examples above, we substituted the x- and y-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for and then solve the equation for . Or, pick a value for and then solve for .
We’ll start by looking at the solutions to the equation that we found in . We can summarize this information in a table of solutions, as shown in .
To find a third solution, we’ll let and solve for .
The ordered pair is a solution to . We will add it to .
We can find more solutions to the equation by substituting in any value of or any value of and solving the resulting equation to get another ordered pair that is a solution. There are infinitely many solutions of this equation.
### Find Solutions to a Linear Equation
To find a solution to a linear equation, you really can pick any number you want to substitute into the equation for or But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.
When the equation is in y-form, with the y by itself on one side of the equation, it is usually easier to choose values of and then solve for .
We have seen how using zero as one value of makes finding the value of easy. When an equation is in standard form, with both the and on the same side of the equation, it is usually easier to first find one solution when find a second solution when , and then find a third solution.
### Key Concepts
1. Sign Patterns of the Quadrants
2. Points on the Axes
3. Solution of a Linear Equation
### Practice Makes Perfect
Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Verify Solutions to an Equation in Two Variables
In the following exercises, which ordered pairs are solutions to the given equations?
Complete a Table of Solutions to a Linear Equation
In the following exercises, complete the table to find solutions to each linear equation.
Find Solutions to a Linear Equation
In the following exercises, find three solutions to each linear equation.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no, I don’t get it. This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Graphs
## Graph Linear Equations in Two Variables
### Recognize the Relationship Between the Solutions of an Equation and its Graph
In the previous section, we found several solutions to the equation . They are listed in . So, the ordered pairs , , and are some solutions to the equation . We can plot these solutions in the rectangular coordinate system as shown in .
Notice how the points line up perfectly? We connect the points with a line to get the graph of the equation . See . Notice the arrows on the ends of each side of the line. These arrows indicate the line continues.
Every point on the line is a solution of the equation. Also, every solution of this equation is a point on this line. Points not on the line are not solutions.
Notice that the point whose coordinates are is on the line shown in . If you substitute and into the equation, you find that it is a solution to the equation.
So the point is a solution to the equation . (The phrase “the point whose coordinates are ” is often shortened to “the point .”)
So is not a solution to the equation . Therefore, the point is not on the line. See . This is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to the equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation .
### Graph a Linear Equation by Plotting Points
There are several methods that can be used to graph a linear equation. The method we used to graph is called plotting points, or the Point–Plotting Method.
The steps to take when graphing a linear equation by plotting points are summarized below.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.
If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between part (a) and part (b) in .
Let’s do another example. This time, we’ll show the last two steps all on one grid.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for . But the math is easier if we make ‘good’ choices for the values of . This way we will avoid fraction answers, which are hard to graph precisely.
So far, all the equations we graphed had given in terms of . Now we’ll graph an equation with and on the same side. Let’s see what happens in the equation . If what is the value of ?
This point has a fraction for the x- coordinate and, while we could graph this point, it is hard to be precise graphing fractions. Remember in the example , we carefully chose values for so as not to graph fractions at all. If we solve the equation for , it will be easier to find three solutions to the equation.
The solutions for , , and are shown in the . The graph is shown in .
Can you locate the point , which we found by letting , on the line?
If you can choose any three points to graph a line, how will you know if your graph matches the one shown in the answers in the book? If the points where the graphs cross the x- and y-axis are the same, the graphs match!
The equation in was written in standard form, with both and on the same side. We solved that equation for in just one step. But for other equations in standard form it is not that easy to solve for , so we will leave them in standard form. We can still find a first point to plot by letting and solving for . We can plot a second point by letting and then solving for . Then we will plot a third point by using some other value for or .
### Graph Vertical and Horizontal Lines
Can we graph an equation with only one variable? Just and no , or just without an ? How will we make a table of values to get the points to plot?
Let’s consider the equation . This equation has only one variable, . The equation says that is always equal to , so its value does not depend on . No matter what is, the value of is always .
So to make a table of values, write in for all the values. Then choose any values for . Since does not depend on , you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .
Plot the points from and connect them with a straight line. Notice in that we have graphed a vertical line.
What if the equation has but no ? Let’s graph the equation . This time the y- value is a constant, so in this equation, does not depend on . Fill in 4 for all the ’s in and then choose any values for . We’ll use 0, 2, and 4 for the x-coordinates.
The graph is a horizontal line passing through the y-axis at 4. See .
The equations for vertical and horizontal lines look very similar to equations like What is the difference between the equations and ?
The equation has both and . The value of depends on the value of . The y-coordinate changes according to the value of . The equation has only one variable. The value of is constant. The y-coordinate is always 4. It does not depend on the value of . See .
Notice, in , the equation gives a slanted line, while gives a horizontal line.
### Key Concepts
1. Graph a Linear Equation by Plotting Points
### Practice Makes Perfect
Recognize the Relationship Between the Solutions of an Equation and its Graph
In the following exercises, for each ordered pair, decide:
ⓐ Is the ordered pair a solution to the equation? ⓑ Is the point on the line?
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal Lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
### Mixed Practice
In the following exercises, graph each equation.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals? |
# Graphs
## Graph with Intercepts
### Identify the x- and y- Intercepts on a Graph
Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x- axis and the y- axis. These points are called the intercepts of the line.
Let’s look at the graphs of the lines in .
First, notice where each of these lines crosses the negative axis. See .
Do you see a pattern?
For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form and is called the . The x- intercept occurs when is zero.
Now, let’s look at the points where these lines cross the y- axis. See .
What is the pattern here?
In each row, the x- coordinate of the point where the line crosses the y- axis is zero. The point where the line crosses the y- axis has the form and is called the y- intercept of the line. The y- intercept occurs when is zero.
### Find the x- and y- Intercepts from an Equation of a Line
Recognizing that the occurs when y is zero and that the y- intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x- intercept, let and solve for x. To find the , let and solve for y.
### Graph a Line Using the Intercepts
To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
The steps to graph a linear equation using the intercepts are summarized below.
### Key Concepts
1. Find the
2. Graph a Linear Equation using the Intercepts
3. Strategy for Choosing the Most Convenient Method to Graph a Line:
### Practice Makes Perfect
Identify the
In the following exercises, find the x- and y- intercepts on each graph.
Find the
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Graphs
## Understand Slope of a Line
When you graph linear equations, you may notice that some lines tilt up as they go from left to right and some lines tilt down. Some lines are very steep and some lines are flatter. What determines whether a line tilts up or down or if it is steep or flat?
In mathematics, the ‘tilt’ of a line is called the slope of the line. The concept of slope has many applications in the real world. The pitch of a roof, grade of a highway, and a ramp for a wheelchair are some examples where you literally see slopes. And when you ride a bicycle, you feel the slope as you pump uphill or coast downhill.
In this section, we will explore the concept of slope.
### Use Geoboards to Model Slope
A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.
We’ll start by stretching a rubber band between two pegs as shown in .
Doesn’t it look like a line?
Now we stretch one part of the rubber band straight up from the left peg and around a third peg to make the sides of a right triangle, as shown in
We carefully make a 90º angle around the third peg, so one of the newly formed lines is vertical and the other is horizontal.
To find the slope of the line, we measure the distance along the vertical and horizontal sides of the triangle. The vertical distance is called the rise and the horizontal distance is called the run, as shown in .
If our geoboard and rubber band look just like the one shown in , the rise is 2. The rubber band goes up 2 units. (Each space is one unit.)
What is the run?
The rubber band goes across 3 units. The run is 3 (see ).
The slope of a line is the ratio of the rise to the run. In mathematics, it is always referred to with the letter .
What is the slope of the line on the geoboard in ?
The line has slope . This means that the line rises 2 units for every 3 units of run.
When we work with geoboards, it is a good idea to get in the habit of starting at a peg on the left and connecting to a peg to the right. If the rise goes up it is positive and if it goes down it is negative. The run will go from left to right and be positive.
Notice that in the slope is positive and in the slope is negative. Do you notice any difference in the two lines shown in (a) and (b)?
We ‘read’ a line from left to right just like we read words in English. As you read from left to right, the line in (a) is going up; it has positive slope. The line in (b) is going down; it has negative slope.
### Use to Find the Slope of a Line from its Graph
Now, we’ll look at some graphs on the -coordinate plane and see how to find their slopes. The method will be very similar to what we just modeled on our geoboards.
To find the slope, we must count out the rise and the run. But where do we start?
We locate two points on the line whose coordinates are integers. We then start with the point on the left and sketch a right triangle, so we can count the rise and run.
In the last two examples, the lines had y-intercepts with integer values, so it was convenient to use the y-intercept as one of the points to find the slope. In the next example, the y-intercept is a fraction. Instead of using that point, we’ll look for two other points whose coordinates are integers. This will make the slope calculations easier.
### Find the Slope of Horizontal and Vertical Lines
Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.
So how do we find the slope of the horizontal line ? One approach would be to graph the horizontal line, find two points on it, and count the rise and the run. Let’s see what happens when we do this.
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The floor of your room is horizontal. Its slope is 0. If you carefully placed a ball on the floor, it would not roll away.
Now, we’ll consider a vertical line, the line.
But we can’t divide by 0. Division by 0 is not defined. So we say that the slope of the vertical line is undefined.
The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.
Remember, we ‘read’ a line from left to right, just like we read written words in English.
### Use the Slope Formula to find the Slope of a Line Between Two Points
Sometimes we’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.
We have seen that an ordered pair gives the coordinates of a point. But when we work with slopes, we use two points. How can the same symbol be used to represent two different points? Mathematicians use subscripts to distinguish the points.
The use of subscripts in math is very much like the use of last name initials in elementary school. Maybe you remember Laura C. and Laura M. in your third grade class?
We will use to identify the first point and to identify the second point.
If we had more than two points, we could use , , and so on.
Let’s see how the rise and run relate to the coordinates of the two points by taking another look at the slope of the line between the points and .
Since we have two points, we will use subscript notation, .
On the graph, we counted the rise of 3 and the run of 5.
Notice that the rise of 3 can be found by subtracting the y-coordinates 6 and 3.
And the run of 5 can be found by subtracting the x-coordinates 7 and 2.
We know . So .
We rewrite the rise and run by putting in the coordinates .
But 6 is , the y-coordinate of the second point and 3 is , the y-coordinate of the first point.
So we can rewrite the slope using subscript notation.
Also, 7 is , the x-coordinate of the second point and 2 is , the x-coordinate of the first point.
So, again, we rewrite the slope using subscript notation.
We’ve shown that is really another version of . We can use this formula to find the slope of a line when we have two points on the line.
### Graph a Line Given a Point and the Slope
Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.
One other method we can use to graph lines is called the point–slope method. We will use this method when we know one point and the slope of the line. We will start by plotting the point and then use the definition of slope to draw the graph of the line.
### Solve Slope Applications
At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.
### Key Concepts
1. Find the Slope of a Line from its Graph using
2. Graph a Line Given a Point and the Slope
3. Slope of a Horizontal Line
4. Slope of a vertical line
### Practice Makes Perfect
Use Geoboards to Model Slope
In the following exercises, find the slope modeled on each geoboard.
In the following exercises, model each slope. Draw a picture to show your results.
Use to find the Slope of a Line from its Graph
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Graphs
## Use the Slope-Intercept Form of an Equation of a Line
### Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.
In Graph Linear Equations in Two Variables, we graphed the line of the equation by plotting points. See . Let’s find the slope of this line.
The red lines show us the rise is 1 and the run is 2. Substituting into the slope formula:
What is the y-intercept of the line? The y-intercept is where the line crosses the y-axis, so y-intercept is . The equation of this line is:
Notice, the line has:
When a linear equation is solved for , the coefficient of the term is the slope and the constant term is the y-coordinate of the y-intercept. We say that the equation is in slope–intercept form.
Sometimes the slope–intercept form is called the “y-form.”
### Identify the Slope and y-Intercept From an Equation of a Line
In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the y-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and y-intercept from the equation of a line.
When an equation of a line is not given in slope–intercept form, our first step will be to solve the equation for .
### Graph a Line Using its Slope and Intercept
Now that we know how to find the slope and y-intercept of a line from its equation, we can graph the line by plotting the y-intercept and then using the slope to find another point.
We have used a grid with and both going from about to 10 for all the equations we’ve graphed so far. Not all linear equations can be graphed on this small grid. Often, especially in applications with real-world data, we’ll need to extend the axes to bigger positive or smaller negative numbers.
Now that we have graphed lines by using the slope and y-intercept, let’s summarize all the methods we have used to graph lines. See .
### Choose the Most Convenient Method to Graph a Line
Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?
While we could plot points, use the slope–intercept form, or find the intercepts for any equation, if we recognize the most convenient way to graph a certain type of equation, our work will be easier. Generally, plotting points is not the most efficient way to graph a line. We saw better methods in sections 4.3, 4.4, and earlier in this section. Let’s look for some patterns to help determine the most convenient method to graph a line.
Here are six equations we graphed in this chapter, and the method we used to graph each of them.
Equations #1 and #2 each have just one variable. Remember, in equations of this form the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.
In equations #3 and #4, both and are on the same side of the equation. These two equations are of the form . We substituted to find the x-intercept and to find the y-intercept, and then found a third point by choosing another value for or .
Equations #5 and #6 are written in slope–intercept form. After identifying the slope and y-intercept from the equation we used them to graph the line.
This leads to the following strategy.
### Graph and Interpret Applications of Slope–Intercept
Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.
Usually when a linear equation models a real-world situation, different letters are used for the variables, instead of x and y. The variable names remind us of what quantities are being measured.
The cost of running some types business has two components—a fixed cost and a variable cost. The fixed cost is always the same regardless of how many units are produced. This is the cost of rent, insurance, equipment, advertising, and other items that must be paid regularly. The variable cost depends on the number of units produced. It is for the material and labor needed to produce each item.
### Use Slopes to Identify Parallel Lines
The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.
We say this more formally in terms of the rectangular coordinate system. Two lines that have the same slope and different y-intercepts are called parallel lines. See .
What about vertical lines? The slope of a vertical line is undefined, so vertical lines don’t fit in the definition above. We say that vertical lines that have different x-intercepts are parallel. See .
Let’s graph the equations and on the same grid. The first equation is already in slope–intercept form: . We solve the second equation for :
Graph the lines.
Notice the lines look parallel. What is the slope of each line? What is the y-intercept of each line?
The slopes of the lines are the same and the y-intercept of each line is different. So we know these lines are parallel.
Since parallel lines have the same slope and different y-intercepts, we can now just look at the slope–intercept form of the equations of lines and decide if the lines are parallel.
### Use Slopes to Identify Perpendicular Lines
Let’s look at the lines whose equations are and , shown in .
These lines lie in the same plane and intersect in right angles. We call these lines perpendicular.
What do you notice about the slopes of these two lines? As we read from left to right, the line rises, so its slope is positive. The line drops from left to right, so it has a negative slope. Does it make sense to you that the slopes of two perpendicular lines will have opposite signs?
If we look at the slope of the first line, , and the slope of the second line, , we can see that they are negative reciprocals of each other. If we multiply them, their product is
This is always true for perpendicular lines and leads us to this definition.
We were able to look at the slope–intercept form of linear equations and determine whether or not the lines were parallel. We can do the same thing for perpendicular lines.
We find the slope–intercept form of the equation, and then see if the slopes are negative reciprocals. If the product of the slopes is , the lines are perpendicular. Perpendicular lines may have the same y-intercepts.
### Key Concepts
1. The slope–intercept form of an equation of a line with slope and y-intercept, is, .
2. Graph a Line Using its Slope and
3. Strategy for Choosing the Most Convenient Method to Graph a Line: Consider the form of the equation.
4. Parallel lines are lines in the same plane that do not intersect.
5. Perpendicular lines are lines in the same plane that form a right angle.
### Practice Makes Perfect
Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation .
Identify the Slope and y-Intercept From an Equation of a Line
In the following exercises, identify the slope and y-intercept of each line.
Graph a Line Using Its Slope and Intercept
In the following exercises, graph the line of each equation using its slope and y-intercept.
Choose the Most Convenient Method to Graph a Line
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.
Use Slopes to Identify Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Graphs
## Find the Equation of a Line
How do online retailers know that ‘you may also like’ a particular item based on something you just ordered? How can economists know how a rise in the minimum wage will affect the unemployment rate? How do medical researchers create drugs to target cancer cells? How can traffic engineers predict the effect on your commuting time of an increase or decrease in gas prices? It’s all mathematics.
You are at an exciting point in your mathematical journey as the mathematics you are studying has interesting applications in the real world.
The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. Data is collected and graphed. If the data points appear to form a straight line, an equation of that line can be used to predict the value of one variable based on the value of the other variable.
To create a mathematical model of a linear relation between two variables, we must be able to find the equation of the line. In this section we will look at several ways to write the equation of a line. The specific method we use will be determined by what information we are given.
### Find an Equation of the Line Given the Slope and y-Intercept
We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, Now, we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.
Sometimes, the slope and intercept need to be determined from the graph.
### Find an Equation of the Line Given the Slope and a Point
Finding an equation of a line using the slope–intercept form of the equation works well when you are given the slope and y-intercept or when you read them off a graph. But what happens when you have another point instead of the y-intercept?
We are going to use the slope formula to derive another form of an equation of the line. Suppose we have a line that has slope and that contains some specific point and some other point, which we will just call . We can write the slope of this line and then change it to a different form.
This format is called the point–slope form of an equation of a line.
We can use the point–slope form of an equation to find an equation of a line when we are given the slope and one point. Then we will rewrite the equation in slope–intercept form. Most applications of linear equations use the the slope–intercept form.
### Find an Equation of the Line Given Two Points
When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.
We have two options so far for finding an equation of a line: slope–intercept or point–slope. Since we will know two points, it will make more sense to use the point–slope form.
But then we need the slope. Can we find the slope with just two points? Yes. Then, once we have the slope, we can use it and one of the given points to find the equation.
We have seen that we can use either the slope–intercept form or the point–slope form to find an equation of a line. Which form we use will depend on the information we are given. This is summarized in .
### Find an Equation of a Line Parallel to a Given Line
Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation.
First let’s look at this graphically.
The graph shows the graph of . We want to graph a line parallel to this line and passing through the point .
We know that parallel lines have the same slope. So the second line will have the same slope as. That slope is. We’ll use the notation to represent the slope of a line parallel to a line with slope . (Notice that the subscript looks like two parallel lines.)
The second line will pass through and have . To graph the line, we start at and count out the rise and run. With (or ), we count out the rise 2 and the run 1. We draw the line.
Do the lines appear parallel? Does the second line pass through ?
Now, let’s see how to do this algebraically.
We can use either the slope–intercept form or the point–slope form to find an equation of a line. Here we know one point and can find the slope. So we will use the point–slope form.
### Find an Equation of a Line Perpendicular to a Given Line
Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point–slope equation, like we did with parallel lines.
The graph shows the graph of . Now, we want to graph a line perpendicular to this line and passing through .
We know that perpendicular lines have slopes that are negative reciprocals. We’ll use the notation to represent the slope of a line perpendicular to a line with slope . (Notice that the subscript ⊥ looks like the right angles made by two perpendicular lines.)
We now know the perpendicular line will pass through with .
To graph the line, we will start at and count out the rise and the run 2. Then we draw the line.
Do the lines appear perpendicular? Does the second line pass through ?
Now, let’s see how to do this algebraically. We can use either the slope–intercept form or the point–slope form to find an equation of a line. In this example we know one point, and can find the slope, so we will use the point–slope form.
In , we used the point–slope form to find the equation. We could have looked at this in a different way.
We want to find a line that is perpendicular to that contains the point . The graph shows us the line and the point .
We know every line perpendicular to a vetical line is horizontal, so we will sketch the horizontal line through .
Do the lines appear perpendicular?
If we look at a few points on this horizontal line, we notice they all have y-coordinates of . So, the equation of the line perpendicular to the vertical line is .
### Key Concepts
1. To Find an Equation of a Line Given the Slope and a Point
2. To Find an Equation of a Line Given Two Points
3. To Write and Equation of a Line
4. To Find an Equation of a Line Parallel to a Given Line
5. To Find an Equation of a Line Perpendicular to a Given Line
### Practice Makes Perfect
Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.
### Mixed Practice
In the following exercises, find the equation of each line. Write the equation in slope–intercept form.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Graphs
## Graphs of Linear Inequalities
### Verify Solutions to an Inequality in Two Variables
We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.
Do you remember that an inequality with one variable had many solutions? The solution to the inequality is any number greater than 3. We showed this on the number line by shading in the number line to the right of 3, and putting an open parenthesis at 3. See .
Similarly, inequalities in two variables have many solutions. Any ordered pair that makes the inequality true when we substitute in the values is a solution of the inequality.
### Recognize the Relation Between the Solutions of an Inequality and its Graph
Now, we will look at how the solutions of an inequality relate to its graph.
Let’s think about the number line in again. The point separated that number line into two parts. On one side of 3 are all the numbers less than 3. On the other side of 3 all the numbers are greater than 3. See .
The solution to is the shaded part of the number line to the right of .
Similarly, the line separates the plane into two regions. On one side of the line are points with . On the other side of the line are the points with . We call the line a boundary line.
For an inequality in one variable, the endpoint is shown with a parenthesis or a bracket depending on whether or not is included in the solution:
Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to indicate whether or not it the line is included in the solution. This is summarized in
Now, let’s take a look at what we found in . We’ll start by graphing the line , and then we’ll plot the five points we tested. See .
In we found that some of the points were solutions to the inequality and some were not.
Which of the points we plotted are solutions to the inequality ? The points and are solutions to the inequality . Notice that they are both on the same side of the boundary line .
The two points and are on the other side of the boundary line , and they are not solutions to the inequality . For those two points, .
What about the point ? Because , the point is a solution to the equation . So the point is on the boundary line.
Let’s take another point on the left side of the boundary line and test whether or not it is a solution to the inequality . The point clearly looks to be to the left of the boundary line, doesn’t it? Is it a solution to the inequality?
Any point you choose on the left side of the boundary line is a solution to the inequality . All points on the left are solutions.
Similarly, all points on the right side of the boundary line, the side with and , are not solutions to . See .
The graph of the inequality is shown in below. The line divides the plane into two regions. The shaded side shows the solutions to the inequality .
The points on the boundary line, those where , are not solutions to the inequality , so the line itself is not part of the solution. We show that by making the line dashed, not solid.
### Graph Linear Inequalities
Now, we’re ready to put all this together to graph linear inequalities.
The steps we take to graph a linear inequality are summarized here.
What if the boundary line goes through the origin? Then we won’t be able to use as a test point. No problem—we’ll just choose some other point that is not on the boundary line.
Some linear inequalities have only one variable. They may have an x but no y, or a y but no x. In these cases, the boundary line will be either a vertical or a horizontal line. Do you remember?
### Key Concepts
1. To Graph a Linear Inequality
### Section Exercises
### Practice Makes Perfect
Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
### Chapter 4 Review Exercises
### Rectangular Coordinate System
Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
Identify Points on a Graph
In the following exercises, name the ordered pair of each point shown in the rectangular coordinate system.
Verify Solutions to an Equation in Two Variables
In the following exercises, which ordered pairs are solutions to the given equations?
Complete a Table of Solutions to a Linear Equation in Two Variables
In the following exercises, complete the table to find solutions to each linear equation.
Find Solutions to a Linear Equation in Two Variables
In the following exercises, find three solutions to each linear equation.
### Graphing Linear Equations
Recognize the Relation Between the Solutions of an Equation and its Graph
In the following exercises, for each ordered pair, decide:
1. ⓐ Is the ordered pair a solution to the equation?
2. ⓑ Is the point on the line?
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
### Graphing with Intercepts
Identify the
In the following exercises, find the x- and y-intercepts.
Find the
In the following exercises, find the intercepts of each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
### Slope of a Line
Use Geoboards to Model Slope
In the following exercises, find the slope modeled on each geoboard.
In the following exercises, model each slope. Draw a picture to show your results.
Use
In the following exercises, find the slope of each line shown.
Find the Slope of Horizontal and Vertical Lines
In the following exercises, find the slope of each line.
Use the Slope Formula to find the Slope of a Line between Two Points
In the following exercises, use the slope formula to find the slope of the line between each pair of points.
Graph a Line Given a Point and the Slope
In the following exercises, graph each line with the given point and slope.
Solve Slope Applications
In the following exercises, solve these slope applications.
### Intercept Form of an Equation of a Line
Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line
In the following exercises, use the graph to find the slope and y-intercept of each line. Compare the values to the equation .
Identify the Slope and y-Intercept from an Equation of a Line
In the following exercises, identify the slope and y-intercept of each line.
Graph a Line Using Its Slope and Intercept
In the following exercises, graph the line of each equation using its slope and y-intercept.
In the following exercises, determine the most convenient method to graph each line.
Graph and Interpret Applications of Slope–Intercept
Use Slopes to Identify Parallel Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel.
Use Slopes to Identify Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are perpendicular.
### Find the Equation of a Line
Find an Equation of the Line Given the Slope and
In the following exercises, find the equation of a line with given slope and y-intercept. Write the equation in slope–intercept form.
In the following exercises, find the equation of the line shown in each graph. Write the equation in slope–intercept form.
Find an Equation of the Line Given the Slope and a Point
In the following exercises, find the equation of a line with given slope and containing the given point. Write the equation in slope–intercept form.
Find an Equation of the Line Given Two Points
In the following exercises, find the equation of a line containing the given points. Write the equation in slope–intercept form.
Find an Equation of a Line Parallel to a Given Line
In the following exercises, find an equation of a line parallel to the given line and contains the given point. Write the equation in slope–intercept form.
Find an Equation of a Line Perpendicular to a Given Line
In the following exercises, find an equation of a line perpendicular to the given line and contains the given point. Write the equation in slope–intercept form.
### Graph Linear Inequalities
Verify Solutions to an Inequality in Two Variables
In the following exercises, determine whether each ordered pair is a solution to the given inequality.
Recognize the Relation Between the Solutions of an Inequality and its Graph
In the following exercises, write the inequality shown by the shaded region.
Graph Linear Inequalities
In the following exercises, graph each linear inequality.
### Practice Test
Find the slope of each line shown.
Graph the line for each of the following equations.
Find the equation of each line. Write the equation in slope–intercept form.
Graph each linear inequality. |
# Systems of Linear Equations
## Introduction
An architect designing a home may have restrictions on both the area and perimeter of the windows because of energy and structural concerns. The length and width chosen for each window would have to satisfy two equations: one for the area and the other for the perimeter. Similarly, a banker may have a fixed amount of money to put into two investment funds. A restaurant owner may want to increase profits, but in order to do that he will need to hire more staff. A job applicant may compare salary and costs of commuting for two job offers.
In this chapter, we will look at methods to solve situations like these using equations with two variables. |
# Systems of Linear Equations
## Solve Systems of Equations by Graphing
### Determine Whether an Ordered Pair is a Solution of a System of Equations
In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.
Now we will work with systems of linear equations, two or more linear equations grouped together.
We will focus our work here on systems of two linear equations in two unknowns. Later, you may solve larger systems of equations.
An example of a system of two linear equations is shown below. We use a brace to show the two equations are grouped together to form a system of equations.
A linear equation in two variables, like 2x + y = 7, has an infinite number of solutions. Its graph is a line. Remember, every point on the line is a solution to the equation and every solution to the equation is a point on the line.
To solve a system of two linear equations, we want to find the values of the variables that are solutions to both equations. In other words, we are looking for the ordered pairs (x, y) that make both equations true. These are called the solutions to a system of equations.
To determine if an ordered pair is a solution to a system of two equations, we substitute the values of the variables into each equation. If the ordered pair makes both equations true, it is a solution to the system.
Let’s consider the system below:
Is the ordered pair a solution?
The ordered pair (2, −1) made both equations true. Therefore (2, −1) is a solution to this system.
Let’s try another ordered pair. Is the ordered pair (3, 2) a solution?
The ordered pair (3, 2) made one equation true, but it made the other equation false. Since it is not a solution to both equations, it is not a solution to this system.
### Solve a System of Linear Equations by Graphing
In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.
The graph of a linear equation is a line. Each point on the line is a solution to the equation. For a system of two equations, we will graph two lines. Then we can see all the points that are solutions to each equation. And, by finding what the lines have in common, we’ll find the solution to the system.
Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions.
Similarly, when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown in :
For the first example of solving a system of linear equations in this section and in the next two sections, we will solve the same system of two linear equations. But we’ll use a different method in each section. After seeing the third method, you’ll decide which method was the most convenient way to solve this system.
The steps to use to solve a system of linear equations by graphing are shown below.
Both equations in were given in slope–intercept form. This made it easy for us to quickly graph the lines. In the next example, we’ll first re-write the equations into slope–intercept form.
Usually when equations are given in standard form, the most convenient way to graph them is by using the intercepts. We’ll do this in .
Do you remember how to graph a linear equation with just one variable? It will be either a vertical or a horizontal line.
In all the systems of linear equations so far, the lines intersected and the solution was one point. In the next two examples, we’ll look at a system of equations that has no solution and at a system of equations that has an infinite number of solutions.
If you write the second equation in in slope-intercept form, you may recognize that the equations have the same slope and same y-intercept.
When we graphed the second line in the last example, we drew it right over the first line. We say the two lines are coincident. Coincident lines have the same slope and same y-intercept.
### Determine the Number of Solutions of a Linear System
There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.
We have seen that two lines in the same plane must either intersect or are parallel. The systems of equations in through all had two intersecting lines. Each system had one solution.
A system with parallel lines, like , has no solution. What happened in ? The equations have coincident lines, and so the system had infinitely many solutions.
We’ll organize these results in below:
Parallel lines have the same slope but different y-intercepts. So, if we write both equations in a system of linear equations in slope–intercept form, we can see how many solutions there will be without graphing! Look at the system we solved in .
The two lines have the same slope but different y-intercepts. They are parallel lines.
shows how to determine the number of solutions of a linear system by looking at the slopes and intercepts.
Let’s take one more look at our equations in that gave us parallel lines.
When both lines were in slope-intercept form we had:
Do you recognize that it is impossible to have a single ordered pair that is a solution to both of those equations?
We call a system of equations like this an inconsistent system. It has no solution.
A system of equations that has at least one solution is called a consistent system.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent equations, they each have their own set of solutions. Intersecting lines and parallel lines are independent.
If two equations are dependent, all the solutions of one equation are also solutions of the other equation. When we graph two dependent equations, we get coincident lines.
Let’s sum this up by looking at the graphs of the three types of systems. See and .
### Solve Applications of Systems of Equations by Graphing
We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.
Step 5 is where we will use the method introduced in this section. We will graph the equations and find the solution.
### Key Concepts
1. To solve a system of linear equations by graphing
2. Determine the number of solutions from the graph of a linear system
3. Determine the number of solutions of a linear system by looking at the slopes and intercepts
4. Determine the number of solutions and how to classify a system of equations
5. Problem Solving Strategy for Systems of Linear Equations
### Practice Makes Perfect
Determine Whether an Ordered Pair is a Solution of a System of Equations. In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing In the following exercises, solve the following systems of equations by graphing.
Determine the Number of Solutions of a Linear System Without graphing the following systems of equations, determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Systems of Linear Equations
## Solving Systems of Equations by Substitution
Solving systems of linear equations by graphing is a good way to visualize the types of solutions that may result. However, there are many cases where solving a system by graphing is inconvenient or imprecise. If the graphs extend beyond the small grid with x and y both between −10 and 10, graphing the lines may be cumbersome. And if the solutions to the system are not integers, it can be hard to read their values precisely from a graph.
In this section, we will solve systems of linear equations by the substitution method.
### Solve a System of Equations by Substitution
We will use the same system we used first for graphing.
We will first solve one of the equations for either x or y. We can choose either equation and solve for either variable—but we’ll try to make a choice that will keep the work easy.
Then we substitute that expression into the other equation. The result is an equation with just one variable—and we know how to solve those!
After we find the value of one variable, we will substitute that value into one of the original equations and solve for the other variable. Finally, we check our solution and make sure it makes both equations true.
We’ll fill in all these steps now in .
If one of the equations in the system is given in slope–intercept form, Step 1 is already done! We’ll see this in .
If the equations are given in standard form, we’ll need to start by solving for one of the variables. In this next example, we’ll solve the first equation for y.
In it was easiest to solve for y in the first equation because it had a coefficient of 1. In it will be easier to solve for x.
When both equations are already solved for the same variable, it is easy to substitute!
Be very careful with the signs in the next example.
In , it will take a little more work to solve one equation for x or y.
Look back at the equations in . Is there any way to recognize that they are the same line?
Let’s see what happens in the next example.
### Solve Applications of Systems of Equations by Substitution
We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.
Some people find setting up word problems with two variables easier than setting them up with just one variable. Choosing the variable names is easier when all you need to do is write down two letters. Think about this in the next example—how would you have done it with just one variable?
In the , we’ll use the formula for the perimeter of a rectangle, P = 2L + 2W.
For we need to remember that the sum of the measures of the angles of a triangle is 180 degrees and that a right triangle has one 90 degree angle.
### Key Concepts
1. Solve a system of equations by substitution
### Practice Makes Perfect
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Systems of Linear Equations
## Solve Systems of Equations by Elimination
We have solved systems of linear equations by graphing and by substitution. Graphing works well when the variable coefficients are small and the solution has integer values. Substitution works well when we can easily solve one equation for one of the variables and not have too many fractions in the resulting expression.
The third method of solving systems of linear equations is called the Elimination Method. When we solved a system by substitution, we started with two equations and two variables and reduced it to one equation with one variable. This is what we’ll do with the elimination method, too, but we’ll have a different way to get there.
### Solve a System of Equations by Elimination
The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.
For any expressions a, b, c, and d,
To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.
Notice how that works when we add these two equations together:
The y’s add to zero and we have one equation with one variable.
Let’s try another one:
This time we don’t see a variable that can be immediately eliminated if we add the equations.
But if we multiply the first equation by −2, we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by −2.
Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations.
Add the equations yourself—the result should be −3y = −6. And that looks easy to solve, doesn’t it? Here is what it would look like.
We’ll do one more:
It doesn’t appear that we can get the coefficients of one variable to be opposites by multiplying one of the equations by a constant, unless we use fractions. So instead, we’ll have to multiply both equations by a constant.
We can make the coefficients of x be opposites if we multiply the first equation by 3 and the second by −4, so we get 12x and −12x.
This gives us these two new equations:
When we add these equations,
the x’s are eliminated and we just have −29y = 58.
Once we get an equation with just one variable, we solve it. Then we substitute that value into one of the original equations to solve for the remaining variable. And, as always, we check our answer to make sure it is a solution to both of the original equations.
Now we’ll see how to use elimination to solve the same system of equations we solved by graphing and by substitution.
The steps are listed below for easy reference.
First we’ll do an example where we can eliminate one variable right away.
In , we will be able to make the coefficients of one variable opposites by multiplying one equation by a constant.
Now we’ll do an example where we need to multiply both equations by constants in order to make the coefficients of one variable opposites.
When the system of equations contains fractions, we will first clear the fractions by multiplying each equation by its LCD.
In the Solving Systems of Equations by Graphing we saw that not all systems of linear equations have a single ordered pair as a solution. When the two equations were really the same line, there were infinitely many solutions. We called that a consistent system. When the two equations described parallel lines, there was no solution. We called that an inconsistent system.
### Solve Applications of Systems of Equations by Elimination
Some applications problems translate directly into equations in standard form, so we will use the elimination method to solve them. As before, we use our Problem Solving Strategy to help us stay focused and organized.
### Choose the Most Convenient Method to Solve a System of Linear Equations
When you will have to solve a system of linear equations in a later math class, you will usually not be told which method to use. You will need to make that decision yourself. So you’ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.
### Key Concepts
1. To Solve a System of Equations by Elimination
### Practice Makes Perfect
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Solve Applications of Systems of Equations by Elimination
In the following exercises, translate to a system of equations and solve.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Systems of Linear Equations
## Solve Applications with Systems of Equations
Previously in this chapter we solved several applications with systems of linear equations. In this section, we’ll look at some specific types of applications that relate two quantities. We’ll translate the words into linear equations, decide which is the most convenient method to use, and then solve them.
We will use our Problem Solving Strategy for Systems of Linear Equations.
### Translate to a System of Equations
Many of the problems we solved in earlier applications related two quantities. Here are two of the examples from the chapter on Math Models.
1. The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
2. A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
In that chapter we translated each situation into one equation using only one variable. Sometimes it was a bit of a challenge figuring out how to name the two quantities, wasn’t it?
Let’s see how we can translate these two problems into a system of equations with two variables. We’ll focus on Steps 1 through 4 of our Problem Solving Strategy.
We’ll do another example where we stop after we write the system of equations.
### Solve Direct Translation Applications
We set up, but did not solve, the systems of equations in and Now we’ll translate a situation to a system of equations and then solve it.
### Solve Geometry Applications
When we learned about Math Models, we solved geometry applications using properties of triangles and rectangles. Now we’ll add to our list some properties of angles.
The measures of two complementary angles add to 90 degrees. The measures of two supplementary angles add to 180 degrees.
If two angles are complementary, we say that one angle is the complement of the other.
If two angles are supplementary, we say that one angle is the supplement of the other.
### Solve Uniform Motion Applications
We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was D = rt where D is the distance travelled, r is the rate, and t is the time.
Our first example of a uniform motion application will be for a situation similar to some we have already seen, but now we can use two variables and two equations.
Many real-world applications of uniform motion arise because of the effects of currents—of water or air—on the actual speed of a vehicle. Cross-country airplane flights in the United States generally take longer going west than going east because of the prevailing wind currents.
Let’s take a look at a boat travelling on a river. Depending on which way the boat is going, the current of the water is either slowing it down or speeding it up.
and show how a river current affects the speed at which a boat is actually travelling. We’ll call the speed of the boat in still water b and the speed of the river current c.
In the boat is going downstream, in the same direction as the river current. The current helps push the boat, so the boat’s actual speed is faster than its speed in still water. The actual speed at which the boat is moving is b + c.
In the boat is going upstream, opposite to the river current. The current is going against the boat, so the boat’s actual speed is slower than its speed in still water. The actual speed of the boat is .
We’ll put some numbers to this situation in .
Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in . A wind current in the same direction as the plane is flying is called a tailwind. A wind current blowing against the direction of the plane is called a headwind.
### Practice Makes Perfect
Translate to a System of Equations
In the following exercises, translate to a system of equations and solve the system.
Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications In the following exercises, translate to a system of equations and solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Systems of Linear Equations
## Solve Mixture Applications with Systems of Equations
### Solve Mixture Applications
When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:
Using one variable meant that we had to relate the number of nickels and the number of dimes. We had to decide if we were going to let n be the number of nickels and then write the number of dimes in terms of n, or if we would let d be the number of dimes and write the number of nickels in terms of d.
Now that we know how to solve systems of equations with two variables, we’ll just let n be the number of nickels and d be the number of dimes. We’ll write one equation based on the total value column, like we did before, and the other equation will come from the number column.
For the first example, we’ll do a ticket problem where the ticket prices are in whole dollars, so we won’t need to use decimals just yet.
In we’ll solve a coin problem. Now that we know how to work with systems of two variables, naming the variables in the ‘number’ column will be easy.
Some mixture applications involve combining foods or drinks. Example situations might include combining raisins and nuts to make a trail mix or using two types of coffee beans to make a blend.
Another application of mixture problems relates to concentrated cleaning supplies, other chemicals, and mixed drinks. The concentration is given as a percent. For example, a 20% concentrated household cleanser means that 20% of the total amount is cleanser, and the rest is water. To make 35 ounces of a 20% concentration, you mix 7 ounces (20% of 35) of the cleanser with 28 ounces of water.
For these kinds of mixture problems, we’ll use percent instead of value for one of the columns in our table.
### Solve Interest Applications
The formula to model interest applications is I = Prt. Interest, I, is the product of the principal, P, the rate, r, and the time, t. In our work here, we will calculate the interest earned in one year, so t will be 1.
We modify the column titles in the mixture table to show the formula for interest, as you’ll see in .
### Key Concepts
1. Table for coin and mixture applications
2. Table for concentration applications
3. Table for interest applications
### Practice Makes Perfect
Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.
### Everyday Math
In the following exercises, translate to a system of equations and solve.
### Writing Exercises
### Self Check
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
After looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Systems of Linear Equations
## Graphing Systems of Linear Inequalities
### Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.
A system of linear inequalities looks like a system of linear equations, but it has inequalities instead of equations. A system of two linear inequalities is shown below.
To solve a system of linear inequalities, we will find values of the variables that are solutions to both inequalities. We solve the system by using the graphs of each inequality and show the solution as a graph. We will find the region on the plane that contains all ordered pairs that make both inequalities true.
To determine if an ordered pair is a solution to a system of two inequalities, we substitute the values of the variables into each inequality. If the ordered pair makes both inequalities true, it is a solution to the system.
### Solve a System of Linear Inequalities by Graphing
The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.
Systems of linear inequalities where the boundary lines are parallel might have no solution. We’ll see this in .
### Solve Applications of Systems of Inequalities
The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.
### Key Concepts
1. To Solve a System of Linear Inequalities by Graphing
### Section Exercises
### Practice Makes Perfect
Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
### Chapter 5 Review Exercises
### Solve Systems of Equations by Graphing
Determine Whether an Ordered Pair is a Solution of a System of Equations.
In the following exercises, determine if the following points are solutions to the given system of equations.
Solve a System of Linear Equations by Graphing
In the following exercises, solve the following systems of equations by graphing.
Determine the Number of Solutions of a Linear System
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve Applications of Systems of Equations by Graphing
### Solve Systems of Equations by Substitution
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve Applications of Systems of Equations by Substitution
In the following exercises, translate to a system of equations and solve.
### Solve Systems of Equations by Elimination
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
Solve Applications of Systems of Equations by Elimination
In the following exercises, translate to a system of equations and solve.
Choose the Most Convenient Method to Solve a System of Linear Equations
In the following exercises, decide whether it would be more convenient to solve the system of equations by substitution or elimination.
### Solve Applications with Systems of Equations
Translate to a System of Equations
In the following exercises, translate to a system of equations. Do not solve the system.
Solve Direct Translation Applications
In the following exercises, translate to a system of equations and solve.
Solve Geometry Applications
In the following exercises, translate to a system of equations and solve.
Solve Uniform Motion Applications
In the following exercises, translate to a system of equations and solve.
### Solve Mixture Applications with Systems of Equations
Solve Mixture Applications
In the following exercises, translate to a system of equations and solve.
Solve Interest Applications
In the following exercises, translate to a system of equations and solve.
### Graphing Systems of Linear Inequalities
Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities
In the following exercises, determine whether each ordered pair is a solution to the system.
Solve a System of Linear Inequalities by Graphing
In the following exercises, solve each system by graphing.
Solve Applications of Systems of Inequalities
In the following exercises, translate to a system of inequalities and solve.
### Practice Test
In the following exercises, solve the following systems by graphing.
In the following exercises, solve each system of equations. Use either substitution or elimination.
In the following exercises, translate to a system of equations and solve. |
# Polynomials
## Introduction
We have seen that the graphs of linear equations are straight lines. Graphs of other types of equations, called polynomial equations, are curves, like the outline of this suspension bridge. Architects use polynomials to design the shape of a bridge like this and to draw the blueprints for it. Engineers use polynomials to calculate the stress on the bridge’s supports to ensure they are strong enough for the intended load. In this chapter, you will explore operations with and properties of polynomials. |
# Polynomials
## Add and Subtract Polynomials
### Identify Polynomials, Monomials, Binomials and Trinomials
You have learned that a term is a constant or the product of a constant and one or more variables. When it is of the form , where is a constant and is a whole number, it is called a monomial. Some examples of monomial are .
A monomial, or two or more monomials combined by addition or subtraction, is a polynomial. Some polynomials have special names, based on the number of terms. A monomial is a polynomial with exactly one term. A binomial has exactly two terms, and a trinomial has exactly three terms. There are no special names for polynomials with more than three terms.
Here are some examples of polynomials.
Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.
### Determine the Degree of Polynomials
The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.
A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0—it has no variable.
Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.
A polynomial is in standard form when the terms of a polynomial are written in descending order of degrees. Get in the habit of writing the term with the highest degree first.
### Add and Subtract Monomials
You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.
Remember that like terms must have the same variables with the same exponents.
### Add and Subtract Polynomials
We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.
### Evaluate a Polynomial for a Given Value
We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.
### Key Concepts
1. Monomials
2. Polynomials
3. Degree of a Polynomial
### Practice Makes Perfect
Identify Polynomials, Monomials, Binomials, and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value
In the following exercises, evaluate each polynomial for the given value.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Polynomials
## Use Multiplication Properties of Exponents
### Simplify Expressions with Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, means to multiply 2 by itself 4 times, so means .
Let’s review the vocabulary for expressions with exponents.
Before we begin working with variable expressions containing exponents, let’s simplify a few expressions involving only numbers.
Notice the similarities and differences in ⓐ and ⓑ ! Why are the answers different? As we follow the order of operations in part ⓐ the parentheses tell us to raise the to the 4th power. In part ⓑ we raise just the 5 to the 4th power and then take the opposite.
### Simplify Expressions Using the Product Property for Exponents
You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.
We’ll derive the properties of exponents by looking for patterns in several examples.
First, we will look at an example that leads to the Product Property.
We write:
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
An example with numbers helps to verify this property.
We can extend the Product Property for Exponents to more than two factors.
### Simplify Expressions Using the Power Property for Exponents
Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.
We write:
We multiplied the exponents. This leads to the Power Property for Exponents.
An example with numbers helps to verify this property.
### Simplify Expressions Using the Product to a Power Property
We will now look at an expression containing a product that is raised to a power. Can you find this pattern?
Notice that each factor was raised to the power and is .
The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.
An example with numbers helps to verify this property:
### Simplify Expressions by Applying Several Properties
We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.
All exponent properties hold true for any real numbers . Right now, we only use whole number exponents.
### Multiply Monomials
Since a monomial is an algebraic expression, we can use the properties of exponents to multiply monomials.
### Key Concepts
1. Exponential Notation
2. Properties of Exponents
### Practice Makes Perfect
Simplify Expressions with Exponents
In the following exercises, simplify each expression with exponents.
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression using the Product Property for Exponents.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression using the Power Property for Exponents.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression using the Product to a Power Property.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises, multiply the monomials.
Mixed Practice
In the following exercises, simplify each expression.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals? |
# Polynomials
## Multiply Polynomials
### Multiply a Polynomial by a Monomial
We have used the Distributive Property to simplify expressions like . You multiplied both terms in the parentheses, , by 2, to get . With this chapter’s new vocabulary, you can say you were multiplying a binomial, , by a monomial, 2.
Multiplying a binomial by a monomial is nothing new for you! Here’s an example:
### Multiply a Binomial by a Binomial
Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.
### Multiply a Binomial by a Binomial Using the Distributive Property
Look at , where we multiplied a binomial by a monomial.
Notice that before combining like terms, you had four terms. You multiplied the two terms of the first binomial by the two terms of the second binomial—four multiplications.
### Multiply a Binomial by a Binomial Using the FOIL Method
Remember that when you multiply a binomial by a binomial you get four terms. Sometimes you can combine like terms to get a trinomial, but sometimes, like in , there are no like terms to combine.
Let’s look at the last example again and pay particular attention to how we got the four terms.
Where did the first term, , come from?
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. The word FOIL is easy to remember and ensures we find all four products.
Let’s look at .
Notice how the terms in third line fit the FOIL pattern.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
We summarize the steps of the FOIL method below. The FOIL method only applies to multiplying binomials, not other polynomials!
When you multiply by the FOIL method, drawing the lines will help your brain focus on the pattern and make it easier to apply.
The final products in the last four examples were trinomials because we could combine the two middle terms. This is not always the case.
Be careful of the exponents in the next example.
### Multiply a Binomial by a Binomial Using the Vertical Method
The FOIL method is usually the quickest method for multiplying two binomials, but it only works for binomials. You can use the Distributive Property to find the product of any two polynomials. Another method that works for all polynomials is the Vertical Method. It is very much like the method you use to multiply whole numbers. Look carefully at this example of multiplying two-digit numbers.
Now we’ll apply this same method to multiply two binomials.
We have now used three methods for multiplying binomials. Be sure to practice each method, and try to decide which one you prefer. The methods are listed here all together, to help you remember them.
### Multiply a Trinomial by a Binomial
We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a trinomial by a binomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.
Now let’s do this same multiplication using the Vertical Method.
We have now seen two methods you can use to multiply a trinomial by a binomial. After you practice each method, you’ll probably find you prefer one way over the other. We list both methods are listed here, for easy reference.
### Key Concepts
1. FOIL Method for Multiplying Two Binomials—To multiply two binomials:
2. Multiplying Two Binomials—To multiply binomials, use the:
3. Multiplying a Trinomial by a Binomial—To multiply a trinomial by a binomial, use the:
### Practice Makes Perfect
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
Mixed Practice
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Polynomials
## Special Products
### Square a Binomial Using the Binomial Squares Pattern
Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and using the methods of the last section, there is less work to do if you learn to use a pattern.
Look at these results. Do you see any patterns?
What about the number of terms? In each example we squared a binomial and the result was a trinomial.
Now look at the in each result. Where did it come from?
The first term is the product of the first terms of each binomial. Since the binomials are identical, it is just the square of the first term!
To get the of the product, .
Where did the come from? Look at the examples and find the pattern.
The last term is the product of the last terms, which is the square of the last term.
To get the .
Finally, look at the . Notice it came from adding the “outer” and the “inner” terms—which are both the same! So the middle term is double the product of the two terms of the binomial.
To get the .
Putting it all together:
To square a binomial:
1. square the first term
2. square the last term
3. double their product
A number example helps verify the pattern.
To multiply usually you’d follow the Order of Operations.
The pattern works!
### Multiply Conjugates Using the Product of Conjugates Pattern
We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.
What do you notice about these pairs of binomials?
Look at the first term of each binomial in each pair.
Notice the first terms are the same in each pair.
Look at the last terms of each binomial in each pair.
Notice the last terms are the same in each pair.
Notice how each pair has one sum and one difference.
A pair of binomials that each have the same first term and the same last term, but one is a sum and one is a difference has a special name. It is called a conjugate pair and is of the form .
There is a nice pattern for finding the product of conjugates. You could, of course, simply FOIL to get the product, but using the pattern makes your work easier.
Let’s look for the pattern by using FOIL to multiply some conjugate pairs.
Each first term is the product of the first terms of the binomials, and since they are identical it is the square of the first term.
The last term came from multiplying the last terms, the square of the last term.
What do you observe about the products?
The product of the two binomials is also a binomial! Most of the products resulting from FOIL have been trinomials.
Why is there no middle term? Notice the two middle terms you get from FOIL combine to 0 in every case, the result of one addition and one subtraction.
The product of conjugates is always of the form . This is called a difference of squares.
This leads to the pattern:
Let’s test this pattern with a numerical example.
Notice, the result is the same!
The binomials in the next example may look backwards – the variable is in the second term. But the two binomials are still conjugates, so we use the same pattern to multiply them.
Now we’ll multiply conjugates that have two variables.
### Recognize and Use the Appropriate Special Product Pattern
We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.
### Key Concepts
1. Binomial Squares Pattern
2. Product of Conjugates Pattern
3. To multiply conjugates:
### Practice Makes Perfect
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Polynomials
## Divide Monomials
### Simplify Expressions Using the Quotient Property for Exponents
Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.
Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.
As before, we’ll try to discover a property by looking at some examples.
Notice, in each case the bases were the same and we subtracted exponents.
When the larger exponent was in the numerator, we were left with factors in the numerator.
When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.
We write:
This leads to the Quotient Property for Exponents.
A couple of examples with numbers may help to verify this property.
Notice the difference in the two previous examples:
1. If we start with more factors in the numerator, we will end up with factors in the numerator.
2. If we start with more factors in the denominator, we will end up with factors in the denominator.
The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.
### Simplify Expressions with an Exponent of Zero
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like . From your earlier work with fractions, you know that:
In words, a number divided by itself is 1. So, , for any , since any number divided by itself is 1.
The Quotient Property for Exponents shows us how to simplify when and when by subtracting exponents. What if ?
Consider , which we know is 1.
Now we will simplify in two ways to lead us to the definition of the zero exponent. In general, for :
We see simplifies to and to 1. So .
In this text, we assume any variable that we raise to the zero power is not zero.
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let’s look at . We can use the product to a power rule to rewrite this expression.
This tells us that any nonzero expression raised to the zero power is one.
### Simplify Expressions Using the Quotient to a Power Property
Now we will look at an example that will lead us to the Quotient to a Power Property.
Notice that the exponent applies to both the numerator and the denominator.
This leads to the Quotient to a Power Property for Exponents.
An example with numbers may help you understand this property:
### Simplify Expressions by Applying Several Properties
We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
### Divide Monomials
You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.
Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.
In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we’ll first find the product of two monomials in the numerator before we simplify the fraction. This follows the order of operations. Remember, a fraction bar is a grouping symbol.
### Key Concepts
1. Quotient Property for Exponents:
2. Zero Exponent
3. Quotient to a Power Property for Exponents:
4. Summary of Exponent Properties
### Practice Makes Perfect
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.
Mixed Practice
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Polynomials
## Divide Polynomials
### Divide a Polynomial by a Monomial
In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.
The method we’ll use to divide a polynomial by a monomial is based on the properties of fraction addition. So we’ll start with an example to review fraction addition.
Now we will do this in reverse to split a single fraction into separate fractions.
We’ll state the fraction addition property here just as you learned it and in reverse.
We use the form on the left to add fractions and we use the form on the right to divide a polynomial by a monomial.
We use this form of fraction addition to divide polynomials by monomials.
Remember that division can be represented as a fraction. When you are asked to divide a polynomial by a monomial and it is not already in fraction form, write a fraction with the polynomial in the numerator and the monomial in the denominator.
When we divide by a negative, we must be extra careful with the signs.
### Divide a Polynomial by a Binomial
To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.
We check division by multiplying the quotient by the divisor.
If we did the division correctly, the product should equal the dividend.
Now we will divide a trinomial by a binomial. As you read through the example, notice how similar the steps are to the numerical example above.
When the divisor has subtraction sign, we must be extra careful when we multiply the partial quotient and then subtract. It may be safer to show that we change the signs and then add.
When we divided 875 by 25, we had no remainder. But sometimes division of numbers does leave a remainder. The same is true when we divide polynomials. In , we’ll have a division that leaves a remainder. We write the remainder as a fraction with the divisor as the denominator.
Look back at the dividends in , , and . The terms were written in descending order of degrees, and there were no missing degrees. The dividend in will be . It is missing an term. We will add in as a placeholder.
In , we will divide by . As we divide we will have to consider the constants as well as the variables.
### Key Concepts
1. Fraction Addition
2. Division of a Polynomial by a Monomial
### Practice Makes Perfect
In the following exercises, divide each polynomial by the monomial.
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals? |
# Polynomials
## Integer Exponents and Scientific Notation
### Use the Definition of a Negative Exponent
We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.
What if we just subtract exponents regardless of which is larger?
Let’s consider .
We subtract the exponent in the denominator from the exponent in the numerator.
We can also simplify by dividing out common factors:
This implies that and it leads us to the definition of a negative exponent.
The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.
Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.
For example, if after simplifying an expression we end up with the expression , we will take one more step and write . The answer is considered to be in simplest form when it has only positive exponents.
In we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.
This leads to the Property of Negative Exponents.
Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.
To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.
This leads us to the Quotient to a Negative Power Property.
When simplifying an expression with exponents, we must be careful to correctly identify the base.
We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.
With negative exponents, the Quotient Rule needs only one form , for . When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.
### Simplify Expressions with Integer Exponents
All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
To simplify a fraction, we use the Quotient Property and subtract the exponents.
### Convert from Decimal Notation to Scientific Notation
Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and . We know that 4,000 means and 0.004 means .
If we write the 1000 as a power of ten in exponential form, we can rewrite these numbers in this way:
When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than 10, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.
It is customary in scientific notation to use as the multiplication sign, even though we avoid using this sign elsewhere in algebra.
If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.
In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.
### Convert Scientific Notation to Decimal Form
How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.
If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.
In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.
The steps are summarized below.
### Multiply and Divide Using Scientific Notation
Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.
### Key Concepts
1. Property of Negative Exponents
2. Quotient to a Negative Exponent
3. To convert a decimal to scientific notation:
4. To convert scientific notation to decimal form:
### Section Exercises
### Practice Makes Perfect
Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply. Write your answer in decimal form.
In the following exercises, divide. Write your answer in decimal form.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?
### Chapter 6 Review Exercises
### Add and Subtract Polynomials
Identify Polynomials, Monomials, Binomials and Trinomials
In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.
Determine the Degree of Polynomials
In the following exercises, determine the degree of each polynomial.
Add and Subtract Monomials
In the following exercises, add or subtract the monomials.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial for a Given Value of the Variable
In the following exercises, evaluate each polynomial for the given value.
### Use Multiplication Properties of Exponents
Simplify Expressions with Exponents
In the following exercises, simplify.
Simplify Expressions Using the Product Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Power Property for Exponents
In the following exercises, simplify each expression.
Simplify Expressions Using the Product to a Power Property
In the following exercises, simplify each expression.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify each expression.
Multiply Monomials
In the following exercises 8, multiply the monomials.
### Multiply Polynomials
Multiply a Polynomial by a Monomial
In the following exercises, multiply.
Multiply a Binomial by a Binomial
In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.
In the following exercises, multiply the binomials. Use any method.
Multiply a Trinomial by a Binomial
In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
### Special Products
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
Multiply Conjugates Using the Product of Conjugates Pattern
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
### Divide Monomials
Simplify Expressions Using the Quotient Property for Exponents
In the following exercises, simplify.
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
Divide Monomials
In the following exercises, divide the monomials.
### Divide Polynomials
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
Divide a Polynomial by a Binomial
In the following exercises, divide each polynomial by the binomial.
### Integer Exponents and Scientific Notation
Use the Definition of a Negative Exponent
In the following exercises, simplify.
Simplify Expressions with Integer Exponents
In the following exercises, simplify.
Convert from Decimal Notation to Scientific Notation
In the following exercises, write each number in scientific notation.
Convert Scientific Notation to Decimal Form
In the following exercises, convert each number to decimal form.
Multiply and Divide Using Scientific Notation
In the following exercises, multiply and write your answer in decimal form.
In the following exercises, divide and write your answer in decimal form.
### Chapter Practice Test
In the following exercises, simplify each expression.
In the following exercises, simplify, and write your answer in decimal form. |
# Factoring
## Introduction
Quadratic expressions may be used to model physical properties of a large bridge, the trajectory of a baseball or rocket, and revenue and profit of a business. By factoring these expressions, specific characteristics of the model can be identified. In this chapter, you will explore the process of factoring expressions and see how factoring is used to solve certain types of equations. |
# Factoring
## Greatest Common Factor and Factor by Grouping
### Find the Greatest Common Factor of Two or More Expressions
Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.
We have learned how to factor numbers to find the least common multiple (LCM) of two or more numbers. Now we will factor expressions and find the greatest common factor of two or more expressions. The method we use is similar to what we used to find the LCM.
First we’ll find the GCF of two numbers.
We summarize the steps we use to find the GCF below.
In the first example, the GCF was a constant. In the next two examples, we will get variables in the greatest common factor.
### Factor the Greatest Common Factor from a Polynomial
Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, 12 as in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:
Now we will start with a product, like , and end with its factors, . To do this we apply the Distributive Property “in reverse.”
We state the Distributive Property here just as you saw it in earlier chapters and “in reverse.”
So how do you use the Distributive Property to factor a polynomial? You just find the GCF of all the terms and write the polynomial as a product!
The expressions in the next example have several factors in common. Remember to write the GCF as the product of all the common factors.
Now we’ll factor the greatest common factor from a trinomial. We start by finding the GCF of all three terms.
When the leading coefficient is negative, we factor the negative out as part of the GCF.
### Factor by Grouping
When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.
(Not all polynomials can be factored. Just like some numbers are prime, some polynomials are prime.)
### Key Concepts
1. Finding the Greatest Common Factor (GCF): To find the GCF of two expressions:
2. Factor the Greatest Common Factor from a Polynomial: To factor a greatest common factor from a polynomial:
3. Factor by Grouping: To factor a polynomial with 4 four or more terms
### Practice Makes Perfect
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
Factor by Grouping
In the following exercises, factor by grouping.
Mixed Practice
In the following exercises, factor.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Factoring
## Factor Trinomials of the Form x2+bx+c
### Factor Trinomials of the Form x2 + bx + c
You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.
To factor the trinomial means to start with the product, , and end with the factors, . You need to think about where each of the terms in the trinomial came from.
The first term came from multiplying the first term in each binomial. So to get in the product, each binomial must start with an x.
The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6.
What two numbers multiply to 6?
The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?
Consider the middle term. It came from adding the outer and inner terms.
So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in —the table will be very helpful when you work with numbers that can be factored in many different ways.
We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of . They are .
You should check this by multiplying.
Looking back, we started with , which is of the form , where and . We factored it into two binomials of the form .
To get the correct factors, we found two numbers m and n whose product is c and sum is b.
Let’s summarize the steps we used to find the factors.
### Factor Trinomials of the Form x2 + bx + c with b Negative, c Positive
In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.
Remember: To get a negative sum and a positive product, the numbers must both be negative.
Again, think about FOIL and where each term in the trinomial came from. Just as before,
1. the first term, , comes from the product of the two first terms in each binomial factor, x and y;
2. the positive last term is the product of the two last terms
3. the negative middle term is the sum of the outer and inner terms.
How do you get a positive product and a negative sum? With two negative numbers.
### Factor Trinomials of the Form with c Negative
Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.
Remember: To get a negative product, the numbers must have different signs.
Let’s make a minor change to the last trinomial and see what effect it has on the factors.
Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.
Let’s summarize the method we just developed to factor trinomials of the form .
### Factor Trinomials of the Form x2 + bxy + cy2
Sometimes you’ll need to factor trinomials of the form with two variables, such as The first term, , is the product of the first terms of the binomial factors, . The in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.
### Key Concepts
1. Factor trinomials of the form
### Practice Makes Perfect
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Mixed Practice
In the following exercises, factor each expression.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all goals? |
# Factoring
## Factor Trinomials of the Form ax2+bx+c
### Recognize a Preliminary Strategy for Factoring
Let’s summarize where we are so far with factoring polynomials. In the first two sections of this chapter, we used three methods of factoring: factoring the GCF, factoring by grouping, and factoring a trinomial by “undoing” FOIL. More methods will follow as you continue in this chapter, as well as later in your studies of algebra.
How will you know when to use each factoring method? As you learn more methods of factoring, how will you know when to apply each method and not get them confused? It will help to organize the factoring methods into a strategy that can guide you to use the correct method.
As you start to factor a polynomial, always ask first, “Is there a greatest common factor?” If there is, factor it first.
The next thing to consider is the type of polynomial. How many terms does it have? Is it a binomial? A trinomial? Or does it have more than three terms?
If it is a trinomial where the leading coefficient is one, , use the “undo FOIL” method.
If it has more than three terms, try the grouping method. This is the only method to use for polynomials of more than three terms.
Some polynomials cannot be factored. They are called “prime.”
Below we summarize the methods we have so far. These are detailed in Choose a strategy to factor polynomials completely.
Use the preliminary strategy to completely factor a polynomial. A polynomial is factored completely if, other than monomials, all of its factors are prime.
### Factor Trinomials of the form ax2 + bx + c with a GCF
Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form .
Remember to always check for a GCF first! Sometimes, after you factor the GCF, the leading coefficient of the trinomial becomes 1 and you can factor it by the methods in the last section. Let’s do a few examples to see how this works.
Watch out for the signs in the next two examples.
In the next example the GCF will include a variable.
### Factor Trinomials using Trial and Error
What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.
Let’s factor the trinomial .
From our earlier work we expect this will factor into two binomials.
We know the first terms of the binomial factors will multiply to give us . The only factors of are . We can place them in the binomials.
Check. Does ?
We know the last terms of the binomials will multiply to 2. Since this trinomial has all positive terms, we only need to consider positive factors. The only factors of 2 are 1 and 2. But we now have two cases to consider as it will make a difference if we write 1, 2, or 2, 1.
Which factors are correct? To decide that, we multiply the inner and outer terms.
Since the middle term of the trinomial is 5x, the factors in the first case will work. Let’s FOIL to check.
Our result of the factoring is:
When the middle term is negative and the last term is positive, the signs in the binomials must both be negative.
When we factor an expression, we always look for a greatest common factor first. If the expression does not have a greatest common factor, there cannot be one in its factors either. This may help us eliminate some of the possible factor combinations.
Don’t forget to look for a GCF first.
### Factor Trinomials using the “ac” Method
Another way to factor trinomials of the form is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!
When the third term of the trinomial is negative, the factors of the third term will have opposite signs.
Don’t forget to look for a common factor!
We can now update the Preliminary Factoring Strategy, as shown in and detailed in Choose a strategy to factor polynomials completely (updated), to include trinomials of the form . Remember, some polynomials are prime and so they cannot be factored.
### Key Concepts
1. Factor Trinomials of the Form See .
2. Factor Trinomials of the Form See .
3. Choose a strategy to factor polynomials completely (updated):
### Practice Makes Perfect
Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Factor Trinomials of the form
In the following exercises, factor completely.
Factor Trinomials Using Trial and Error
In the following exercises, factor.
Factor Trinomials using the ‘ac’ Method
In the following exercises, factor.
Mixed Practice
In the following exercises, factor.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Factoring
## Factor Special Products
The strategy for factoring we developed in the last section will guide you as you factor most binomials, trinomials, and polynomials with more than three terms. We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.
### Factor Perfect Square Trinomials
Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.
The first term is the square of the first term of the binomial and the last term is the square of the last. The middle term is twice the product of the two terms of the binomial.
The trinomial 9x2 + 24 +16 is called a perfect square trinomial. It is the square of the binomial 3x+4.
We’ll repeat the Binomial Squares Pattern here to use as a reference in factoring.
When you square a binomial, the product is a perfect square trinomial. In this chapter, you are learning to factor—now, you will start with a perfect square trinomial and factor it into its prime factors.
You could factor this trinomial using the methods described in the last section, since it is of the form ax2 + bx + c. But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.
Here is the pattern—the reverse of the binomial squares pattern.
To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, . Next check that the last term is a perfect square, . Then check the middle term—is it twice the product, 2ab? If everything checks, you can easily write the factors.
The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern , which factors to .
The steps are summarized here.
We’ll work one now where the middle term is negative.
The next example will be a perfect square trinomial with two variables.
Remember the very first step in our Strategy for Factoring Polynomials? It was to ask “is there a greatest common factor?” and, if there was, you factor the GCF before going any further. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.
### Factor Differences of Squares
The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:
Remember, when you multiply conjugate binomials, the middle terms of the product add to 0. All you have left is a binomial, the difference of squares.
Multiplying conjugates is the only way to get a binomial from the product of two binomials.
To factor, we will use the product pattern “in reverse” to factor the difference of squares. A difference of squares factors to a product of conjugates.
Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.
It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression is prime!
Don’t forget that 1 is a perfect square. We’ll need to use that fact in the next example.
The binomial in the next example may look “backwards,” but it’s still the difference of squares.
To completely factor the binomial in the next example, we’ll factor a difference of squares twice!
As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.
### Factor Sums and Differences of Cubes
There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.
We’ll check the first pattern and leave the second to you.
The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.
The trinomial factor in the sum and difference of cubes pattern cannot be factored.
It can be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in .
Be careful to use the correct signs in the factors of the sum and difference of cubes.
In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.
### Key Concepts
1. Factor perfect square trinomials See .
2. Factor differences of squares See .
3. Factor sum and difference of cubes To factor the sum or difference of cubes: See .
### Practice Makes Perfect
Factor Perfect Square Trinomials
In the following exercises, factor.
Factor Differences of Squares
In the following exercises, factor.
Factor Sums and Differences of Cubes
In the following exercises, factor.
Mixed Practice
In the following exercises, factor.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Factoring
## General Strategy for Factoring Polynomials
### Recognize and Use the Appropriate Method to Factor a Polynomial Completely
You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. outlines a strategy you should use when factoring polynomials.
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
### Key Concepts
1. General Strategy for Factoring Polynomials See
2. How to Factor Polynomials
### Practice Makes Perfect
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Factoring
## Quadratic Equations
We have already solved linear equations, equations of the form . In linear equations, the variables have no exponents. Quadratic equations are equations in which the variable is squared. Listed below are some examples of quadratic equations:
The last equation doesn’t appear to have the variable squared, but when we simplify the expression on the left we will get .
The general form of a quadratic equation is .
To solve quadratic equations we need methods different than the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.
### Solve Quadratic Equations Using the Zero Product Property
We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.
We will now use the Zero Product Property, to solve a quadratic equation.
We usually will do a little more work than we did in this last example to solve the linear equations that result from using the Zero Product Property.
Notice when we checked the solutions that each of them made just one factor equal to zero. But the product was zero for both solutions.
It may appear that there is only one factor in the next example. Remember, however, that means .
### Solve Quadratic Equations by Factoring
Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we must be sure to start with the quadratic equation in standard form, . Then we can factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solving quadratic equations by factoring will make use of all the factoring techniques you have learned in this chapter! Do you recognize the special product pattern in the next example?
The left side in the next example is factored, but the right side is not zero. In order to use the Zero Product Property, one side of the equation must be zero. We’ll multiply the factors and then write the equation in standard form.
The Zero Product Property also applies to the product of three or more factors. If the product is zero, at least one of the factors must be zero. We can solve some equations of degree more than two by using the Zero Product Property, just like we solved quadratic equations.
When we factor the quadratic equation in the next example we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.
### Solve Applications Modeled by Quadratic Equations
The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.
We will start with a number problem to get practice translating words into a quadratic equation.
Were you surprised by the pair of negative integers that is one of the solutions to the previous example? The product of the two positive integers and the product of the two negative integers both give 132.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In an earlier chapter, we used the Pythagorean Theorem . It gave the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.
### Key Concepts
1. Zero Product Property If , then either or or both. See .
2. Solve a quadratic equation by factoring To solve a quadratic equation by factoring: See .
3. Use a problem solving strategy to solve word problems See .
### Section Exercises
### Practice Makes Perfect
Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next section? Why or why not?
### Chapter 7 Review Exercises
### 7.1 Greatest Common Factor and Factor by Grouping
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
Factor by Grouping
In the following exercises, factor by grouping.
### 7.2 Factor Trinomials of the form
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form .
Factor Trinomials of the Form
In the following examples, factor each trinomial of the form .
### 7.3 Factoring Trinomials of the form
Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
Factor Trinomials of the Form
In the following exercises, factor completely.
Factor Trinomials Using the “ac” Method
In the following exercises, factor.
Factor Trinomials with a GCF Using the “ac” Method
In the following exercises, factor.
### 7.4 Factoring Special Products
Factor Perfect Square Trinomials
In the following exercises, factor.
Factor Differences of Squares
In the following exercises, factor.
Factor Sums and Differences of Cubes
In the following exercises, factor.
### 7.5 General Strategy for Factoring Polynomials
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
### 7.6 Quadratic Equations
Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
### Practice Test
In the following exercises, find the Greatest Common Factor in each expression.
In the following exercises, factor completely.
In the following exercises, solve. |
# Rational Expressions and Equations
## Introduction
Like rowing a boat, riding a bicycle is a situation in which going in one direction, downhill, is easy, but going in the opposite direction, uphill, can be more work. The trip to reach a destination may be quick, but the return trip whether upstream or uphill will take longer.
Rational equations are used to model situations like these. In this chapter, we will work with rational expressions, solve rational equations, and use them to solve problems in a variety of applications. |
# Rational Expressions and Equations
## Simplify Rational Expressions
In Chapter 1, we reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers, and the denominator is not zero.
In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call these rational expressions.
Remember, division by 0 is undefined.
Here are some examples of rational expressions:
Notice that the first rational expression listed above, is just a fraction. Since a constant is a polynomial with degree zero, the ratio of two constants is a rational expression, provided the denominator is not zero.
We will perform the same operations with rational expressions that we do with fractions. We will simplify, add, subtract, multiply, divide, and use them in applications.
### Determine the Values for Which a Rational Expression is Undefined
When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.
If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
So before we begin any operation with a rational expression, we examine it first to find the values that would make the denominator zero. That way, when we solve a rational equation for example, we will know whether the algebraic solutions we find are allowed or not.
### Evaluate Rational Expressions
To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.
Remember that a fraction is simplified when it has no common factors, other than 1, in its numerator and denominator. When we evaluate a rational expression, we make sure to simplify the resulting fraction.
### Simplify Rational Expressions
Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is simplified if it has no common factors, other than 1, in its numerator and denominator.
For example:
1. is simplified because there are no common factors of 2 and 3.
2. is not simplified because x is a common factor of 2x and 3x.
We use the Equivalent Fractions Property to simplify numerical fractions. We restate it here as we will also use it to simplify rational expressions.
Notice that in the Equivalent Fractions Property, the values that would make the denominators zero are specifically disallowed. We see clearly stated. Every time we write a rational expression, we should make a similar statement disallowing values that would make a denominator zero. However, to let us focus on the work at hand, we will omit writing it in the examples.
Let’s start by reviewing how we simplify numerical fractions.
Throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, and .
To simplify rational expressions we first write the numerator and denominator in factored form. Then we remove the common factors using the Equivalent Fractions Property.
Be very careful as you remove common factors. Factors are multiplied to make a product. You can remove a factor from a product. You cannot remove a term from a sum.
Note that removing the x’s from would be like cancelling the 2’s in the fraction !
We now summarize the steps you should follow to simplify rational expressions.
Usually, we leave the simplified rational expression in factored form. This way it is easy to check that we have removed all the common factors!
We’ll use the methods we covered in Factoring to factor the polynomials in the numerators and denominators in the following examples.
### Simplify Rational Expressions with Opposite Factors
Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say . We know this fraction simplifies to . We also recognize that the numerator and denominator are opposites.
In Foundations, we introduced opposite notation: the opposite of is . We remember, too, that .
We simplify the fraction , whose numerator and denominator are opposites, in this way:
So, in the same way, we can simplify the fraction :
But the opposite of could be written differently:
This means the fraction simplifies to .
In general, we could write the opposite of as . So the rational expression simplifies to .
We will use this property to simplify rational expressions that contain opposites in their numerators and denominators.
Remember, the first step in simplifying a rational expression is to factor the numerator and denominator completely.
### Key Concepts
1. Determine the Values for Which a Rational Expression is Undefined
2. Simplified Rational Expression
3. Simplify a Rational Expression
4. Opposites in a Rational Expression
### Practice Makes Perfect
In the following exercises, determine the values for which the rational expression is undefined.
Evaluate Rational Expressions
In the following exercises, evaluate the rational expression for the given values.
Simplify Rational Expressions
In the following exercises, simplify.
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify each rational expression.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential - every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Rational Expressions and Equations
## Multiply and Divide Rational Expressions
### Multiply Rational Expressions
To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.
We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.
Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, and .
### Divide Rational Expressions
To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.
Remember, the reciprocal of is . To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.
Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.
Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide , we first write 4 as a fraction so that we can find its reciprocal.
We do the same thing when we divide rational expressions.
Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.
If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.
### Key Concepts
1. Multiplication of Rational Expressions
2. Multiply a Rational Expression
3. Division of Rational Expressions
4. Divide Rational Expressions
### Practice Makes Perfect
Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Rational Expressions and Equations
## Add and Subtract Rational Expressions with a Common Denominator
### Add Rational Expressions with a Common Denominator
What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.
It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.
We will add two numerical fractions first, to remind us of how this is done.
Remember, we do not allow values that would make the denominator zero. What value of should be excluded in the next example?
### Subtract Rational Expressions with a Common Denominator
To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.
We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.
Be careful of the signs when you subtract a binomial!
### Add and Subtract Rational Expressions whose Denominators are Opposites
When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by .
Let’s see how this works.
### Key Concepts
1. Rational Expression Addition
2. Rational Expression Subtraction
### Practice Makes Perfect
Add Rational Expressions with a Common Denominator
In the following exercises, add.
Subtract Rational Expressions with a Common Denominator
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add.
In the following exercises, subtract.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Rational Expressions and Equations
## Add and Subtract Rational Expressions with Unlike Denominators
### Find the Least Common Denominator of Rational Expressions
When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.
Let’s look at the example from Foundations. Since the denominators are not the same, the first step was to find the least common denominator (LCD). Remember, the LCD is the least common multiple of the denominators. It is the smallest number we can use as a common denominator.
To find the LCD of 12 and 18, we factored each number into primes, lining up any common primes in columns. Then we “brought down” one prime from each column. Finally, we multiplied the factors to find the LCD.
We do the same thing for rational expressions. However, we leave the LCD in factored form.
### Find Equivalent Rational Expressions
When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.
We will do the same thing for rational expressions.
### Add Rational Expressions with Different Denominators
Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.
Now we will add rational expressions whose denominators are monomials.
Now we are ready to tackle polynomial denominators.
The steps to use to add rational expressions are summarized in the following procedure box.
Avoid the temptation to simplify too soon! In the example above, we must leave the first rational expression as to be able to add it to . Simplify only after you have combined the numerators.
### Subtract Rational Expressions with Different Denominators
The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.
The steps to take to subtract rational expressions are listed below.
There are lots of negative signs in the next example. Be extra careful!
When one expression is not in fraction form, we can write it as a fraction with denominator 1.
We follow the same steps as before to find the LCD when we have more than two rational expressions. In the next example we will start by factoring all three denominators to find their LCD.
### Key Concepts
1. Find the Least Common Denominator of Rational Expressions
2. Add or Subtract Rational Expressions
### Practice Makes Perfect
In the following exercises, find the LCD.
In the following exercises, write as equivalent rational expressions with the given LCD.
In the following exercises, add.
In the following exercises, subtract.
In the following exercises, add and subtract.
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Rational Expressions and Equations
## Simplify Complex Rational Expressions
Complex fractions are fractions in which the numerator or denominator contains a fraction. In Chapter 1 we simplified complex fractions like these:
In this section we will simplify complex rational expressions, which are rational expressions with rational expressions in the numerator or denominator.
Here are a few complex rational expressions:
Remember, we always exclude values that would make any denominator zero.
We will use two methods to simplify complex rational expressions.
### Simplify a Complex Rational Expression by Writing it as Division
We have already seen this complex rational expression earlier in this chapter.
We noted that fraction bars tell us to divide, so rewrote it as the division problem
Then we multiplied the first rational expression by the reciprocal of the second, just like we do when we divide two fractions.
This is one method to simplify rational expressions. We write it as if we were dividing two fractions.
Fraction bars act as grouping symbols. So to follow the Order of Operations, we simplify the numerator and denominator as much as possible before we can do the division.
### Simplify a Complex Rational Expression by Using the LCD
We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.
Let’s look at the complex rational expression we simplified one way in . We will simplify it here by multiplying the numerator and denominator by the LCD. When we multiply by we are multiplying by 1, so the value stays the same.
Be sure to start by factoring all the denominators so you can find the LCD.
### Key Concepts
1. To Simplify a Rational Expression by Writing it as Division
2. To Simplify a Complex Rational Expression by Using the LCD
### Practice Makes Perfect
Simplify a Complex Rational Expression by Writing It as Division
In the following exercises, simplify.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify.
Simplify
In the following exercises, use either method.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Rational Expressions and Equations
## Solve Rational Equations
After defining the terms expression and equation early in Foundations, we have used them throughout this book. We have simplified many kinds of expressions and solved many kinds of equations. We have simplified many rational expressions so far in this chapter. Now we will solve rational equations.
The definition of a rational equation is similar to the definition of equation we used in Foundations.
You must make sure to know the difference between rational expressions and rational equations. The equation contains an equal sign.
### Solve Rational Equations
We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.
Here is an example we did when we worked with linear equations:
We will use the same strategy to solve rational equations. We will multiply both sides of the equation by the LCD. Then we will have an equation that does not contain rational expressions and thus is much easier for us to solve.
But because the original equation may have a variable in a denominator we must be careful that we don’t end up with a solution that would make a denominator equal to zero.
So before we begin solving a rational equation, we examine it first to find the values that would make any denominators zero. That way, when we solve a rational equation we will know if there are any algebraic solutions we must discard.
An algebraic solution to a rational equation that would cause any of the rational expressions to be undefined is called an extraneous solution.
We note any possible extraneous solutions, c, by writing next to the equation.
The steps of this method are shown below.
We always start by noting the values that would cause any denominators to be zero.
When one of the denominators is a quadratic, remember to factor it first to find the LCD.
The equation we solved in had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. Some equations have no solution.
### Solve a Rational Equation for a Specific Variable
When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.
We’ll start with a formula relating distance, rate, and time. We have used it many times before, but not usually in this form.
uses the formula for slope that we used to get the point-slope form of an equation of a line.
Be sure to follow all the steps in . It may look like a very simple formula, but we cannot solve it instantly for either denominator.
### Key Concepts
1. Strategy to Solve Equations with Rational Expressions
### Practice Makes Perfect
Solve Rational Equations
In the following exercises, solve.
Solve a Rational Equation for a Specific Variable
In the following exercises, solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Rational Expressions and Equations
## Solve Proportion and Similar Figure Applications
### Solve Proportions
When two rational expressions are equal, the equation relating them is called a proportion.
The equation is a proportion because the two fractions are equal. The proportion is read “1 is to 2 as 4 is to 8.”
Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.
Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:
We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.
Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.
So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.
Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.
When we work with proportions, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?
To solve applications with proportions, we will follow our usual strategy for solving applications. But when we set up the proportion, we must make sure to have the units correct—the units in the numerators must match and the units in the denominators must match.
In the example above, we related the number of pesos to the number of dollars by using a proportion. We could say the number of pesos is proportional to the number of dollars. If two quantities are related by a proportion, we say that they are proportional.
### Solve Similar Figure Applications
When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.
For example, the two triangles in are similar. Each side of is 4 times the length of the corresponding side of .
This is summed up in the Property of Similar Triangles.
To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.
The next example shows how similar triangles are used with maps.
We can use similar figures to find heights that we cannot directly measure.
### Key Concepts
1. Property of Similar Triangles
2. Problem Solving Strategy for Geometry Applications
### Practice Makes Perfect
Solve Proportions
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, is similar to . Find the length of the indicated side.
In the following exercises, is similar to .
In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.
In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Rational Expressions and Equations
## Solve Uniform Motion and Work Applications
### Solve Uniform Motion Applications
We have solved uniform motion problems using the formula in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula assumes we know r and t and use them to find D. If we know D and r and need to find t, we would solve the equation for t and get the formula .
We have also explained how flying with or against a current affects the speed of a vehicle. We will revisit that idea in the next example.
In the next example, we will know the total time resulting from travelling different distances at different speeds.
Once again, we will use the uniform motion formula solved for the variable t.
### Solve Work Applications
Suppose Pete can paint a room in 10 hours. If he works at a steady pace, in 1 hour he would paint of the room. If Alicia would take 8 hours to paint the same room, then in 1 hour she would paint of the room. How long would it take Pete and Alicia to paint the room if they worked together (and didn’t interfere with each other’s progress)?
This is a typical ‘work’ application. There are three quantities involved here – the time it would take each of the two people to do the job alone and the time it would take for them to do the job together.
Let’s get back to Pete and Alicia painting the room. We will let t be the number of hours it would take them to paint the room together. So in 1 hour working together they have completed of the job.
In one hour Pete did of the job. Alicia did of the job. And together they did of the job.
We can model this with the word equation and then translate to a rational equation. To find the time it would take them if they worked together, we solve for t.
Keep in mind, it should take less time for two people to complete a job working together than for either person to do it alone.
### Practice Makes Perfect
Solve Uniform Motion Applications
In the following exercises, solve uniform motion applications
Solve Work Applications
In the following exercises, solve work applications.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Rational Expressions and Equations
## Use Direct and Inverse Variation
When two quantities are related by a proportion, we say they are proportional to each other. Another way to express this relation is to talk about the variation of the two quantities. We will discuss direct variation and inverse variation in this section.
### Solve Direct Variation Problems
Lindsay gets paid $15 per hour at her job. If we let s be her salary and h be the number of hours she has worked, we could model this situation with the equation
Lindsay’s salary is the product of a constant, 15, and the number of hours she works. We say that Lindsay’s salary varies directly with the number of hours she works. Two variables vary directly if one is the product of a constant and the other.
In applications using direct variation, generally we will know values of one pair of the variables and will be asked to find the equation that relates x and y. Then we can use that equation to find values of y for other values of x.
We’ll list the steps below.
Now we’ll solve a few applications of direct variation.
In the previous example, the variables c and m were named in the problem. Usually that is not the case. We will have to name the variables in the next example as part of the solution, just like we do in most applied problems.
In some situations, one variable varies directly with the square of the other variable. When that happens, the equation of direct variation is . We solve these applications just as we did the previous ones, by substituting the given values into the equation to solve for k.
### Solve Inverse Variation Problems
Many applications involve two variable that vary inversely. As one variable increases, the other decreases. The equation that relates them is .
The word ‘inverse’ in inverse variation refers to the multiplicative inverse. The multiplicative inverse of x is .
We solve inverse variation problems in the same way we solved direct variation problems. Only the general form of the equation has changed. We will copy the procedure box here and just change ‘direct’ to ‘inverse’.
### Section Exercises
### Practice Makes Perfect
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
Write an inverse variation equation to solve the following problems.
### Mixed Practice
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?
### Chapter 8 Review Exercises
### Simplify Rational Expressions
Determine the Values for Which a Rational Expression is Undefined
In the following exercises, determine the values for which the rational expression is undefined.
Evaluate Rational Expressions
In the following exercises, evaluate the rational expressions for the given values.
Simplify Rational Expressions
In the following exercises, simplify.
Simplify Rational Expressions with Opposite Factors
In the following exercises, simplify.
### Multiply and Divide Rational Expressions
Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.
### Add and Subtract Rational Expressions with a Common Denominator
Add Rational Expressions with a Common Denominator
In the following exercises, add.
Subtract Rational Expressions with a Common Denominator
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add and subtract.
### Add and Subtract Rational Expressions With Unlike Denominators
Find the Least Common Denominator of Rational Expressions
In the following exercises, find the LCD.
Find Equivalent Rational Expressions
In the following exercises, rewrite as equivalent rational expressions with the given denominator.
Add Rational Expressions with Different Denominators
In the following exercises, add.
Subtract Rational Expressions with Different Denominators
In the following exercises, subtract and add.
### Simplify Complex Rational Expressions
Simplify a Complex Rational Expression by Writing it as Division
In the following exercises, simplify.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify.
### Solve Rational Equations
Solve Rational Equations
In the following exercises, solve.
Solve a Rational Equation for a Specific Variable
In the following exercises, solve for the indicated variable.
### Solve Proportion and Similar Figure Applications Similarity
Solve Proportions
In the following exercises, solve.
In the following exercises, solve using proportions.
Solve Similar Figure Applications
In the following exercises, solve.
### Solve Uniform Motion and Work Applications Problems
Solve Uniform Motion Applications
In the following exercises, solve.
Solve Work Applications
In the following exercises, solve.
### Use Direct and Inverse Variation
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
### Practice Test
In the following exercises, simplify.
In the following exercises, perform the indicated operation and simplify.
In the following exercises, solve each equation.
In the following exercises, solve. |
# Roots and Radicals
## Introduction
Suppose a stone falls from the edge of a cliff. The number of feet the stone has dropped after seconds can be found by multiplying 16 times the square of . But to calculate the number of seconds it would take the stone to hit the land below, we need to use a square root. In this chapter, we will introduce and apply the properties of square roots, and extend these concepts to higher order roots and rational exponents. |
# Roots and Radicals
## Simplify and Use Square Roots
### Simplify Expressions with Square Roots
Remember that when a number is multiplied by itself, we write and read it “n squared.” For example, reads as “15 squared,” and 225 is called the square of 15, since .
Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 225 is the square of 15, we can also say that 15 is a square root of 225. A number whose square is is called a square root of .
Notice also, so is also a square root of 225. Therefore, both 15 and are square roots of 225.
So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? The radical sign, , denotes the positive square root. The positive square root is also called the principal square root.
We also use the radical sign for the square root of zero. Because , . Notice that zero has only one square root.
Since 15 is the positive square root of 225, we write . Fill in to make a table of square roots you can refer to as you work this chapter.
We know that every positive number has two square roots and the radical sign indicates the positive one. We write . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, .
When using the order of operations to simplify an expression that has square roots, we treat the radical as a grouping symbol.
### Estimate Square Roots
So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at below.
The square roots of numbers between 4 and 9 must be between the two consecutive whole numbers 2 and 3, and they are not whole numbers. Based on the pattern in the table above, we could say that must be between 2 and 3. Using inequality symbols, we write:
### Approximate Square Roots
There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the key on your calculator. You will use this key to approximate square roots.
When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact square root. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately.’
Suppose your calculator has a 10-digit display. You would see that
If we wanted to round to two decimal places, we would say
How do we know these values are approximations and not the exact values? Look at what happens when we square them:
Their squares are close to 5, but are not exactly equal to 5.
Using the square root key on a calculator and then rounding to two decimal places, we can find:
### Simplify Variable Expressions with Square Roots
What if we have to find a square root of an expression with a variable? Consider . Can you think of an expression whose square is ?
When we use the radical sign to take the square root of a variable expression, we should specify that to make sure we get the principal square root.
However, in this chapter we will assume that each variable in a square-root expression represents a non-negative number and so we will not write next to every radical.
What about square roots of higher powers of variables? Think about the Power Property of Exponents we used in Chapter 6.
If we square , the exponent will become .
How does this help us take square roots? Let’s look at a few:
### Key Concepts
1. Note that the square root of a negative number is not a real number.
2. Every positive number has two square roots, one positive and one negative. The positive square root of a positive number is the principal square root.
3. We can estimate square roots using nearby perfect squares.
4. We can approximate square roots using a calculator.
5. When we use the radical sign to take the square root of a variable expression, we should specify that to make sure we get the principal square root.
### Practice Makes Perfect
Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Roots and Radicals
## Simplify Square Roots
In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use .
But what if we want to estimate ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.
A square root is considered simplified if its radicand contains no perfect square factors.
So is simplified. But is not simplified, because 16 is a perfect square factor of 32.
### Use the Product Property to Simplify Square Roots
The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that . The corresponding property of square roots says that .
We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in .
Notice in the previous example that the simplified form of is , which is the product of an integer and a square root. We always write the integer in front of the square root.
We could use the simplified form to estimate . We know 5 is between 2 and 3, and is . So is between 20 and 30.
The next example is much like the previous examples, but with variables.
We follow the same procedure when there is a coefficient in the radical, too.
In the next example both the constant and the variable have perfect square factors.
We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify we must simplify each square root separately first, then add to get the sum of 17.
The expression cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.
In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.
The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.
### Use the Quotient Property to Simplify Square Roots
Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.
If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, .
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.
Be sure to simplify the fraction in the radicand first, if possible.
### Key Concepts
1. Simplified Square Root is considered simplified if has no perfect-square factors.
2. Product Property of Square Roots If a, b are non-negative real numbers, then
3. Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:
4. Quotient Property of Square Roots If a, b are non-negative real numbers and , then
5. Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:
### Practice Makes Perfect
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Roots and Radicals
## Add and Subtract Square Roots
We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify in this way:
So if we have to add , we must not combine them into one radical.
Trying to add square roots with different radicands is like trying to add unlike terms.
Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.
We add and subtract like square roots in the same way we add and subtract like terms. We know that is . Similarly we add and the result is
### Add and Subtract Like Square Roots
Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.
When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.
### Add and Subtract Square Roots that Need Simplification
Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.
Just like we use the Associative Property of Multiplication to simplify and get , we can simplify and get . We will use the Associative Property to do this in the next example.
In the next example, we will remove constant and variable factors from the square roots.
### Key Concepts
1. To add or subtract like square roots, add or subtract the coefficients and keep the like square root.
2. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.
### Practice Makes Perfect
Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
Mixed Practice
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Roots and Radicals
## Multiply Square Roots
### Multiply Square Roots
We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says
We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.
Remember, we assume all variables are greater than or equal to zero.
We will rewrite the Product Property of Square Roots so we see both ways together.
So we can multiply in this way:
Sometimes the product gives us a perfect square:
Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.
Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply we multiply the coefficients together and then the variables. The result is . Keep this in mind as you do these examples.
When we have to multiply square roots, we first find the product and then remove any perfect square factors.
The results of the previous example lead us to this property.
By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.
### Use Polynomial Multiplication to Multiply Square Roots
In the next few examples, we will use the Distributive Property to multiply expressions with square roots.
We will first distribute and then simplify the square roots when possible.
When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.
Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.
We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.
In the next two examples, we will find the product of conjugates.
### Key Concepts
1. Product Property of Square Roots If a, b are nonnegative real numbers, then
2. Special formulas for multiplying binomials and conjugates:
3. The FOIL method can be used to multiply binomials containing radicals.
### Practice Makes Perfect
Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Roots and Radicals
## Divide Square Roots
### Divide Square Roots
We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.
We have used the Quotient Property of Square Roots to simplify square roots of fractions. The Quotient Property of Square Roots says
Sometimes we will need to use the Quotient Property of Square Roots ‘in reverse’ to simplify a fraction with square roots.
We will rewrite the Quotient Property of Square Roots so we see both ways together. Remember: we assume all variables are greater than or equal to zero so that their square roots are real numbers.
We will use the Quotient Property of Square Roots ‘in reverse’ when the fraction we start with is the quotient of two square roots, and neither radicand is a perfect square. When we write the fraction in a single square root, we may find common factors in the numerator and denominator.
We will use the Quotient Property for Exponents, , when we have variables with exponents in the radicands.
### Rationalize a One Term Denominator
Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.
If someone needed to approximate a fraction with a square root in the denominator, it meant doing long division with a five decimal-place divisor. This was a very cumbersome process.
For this reason, a process called rationalizing the denominator was developed. A fraction with a radical in the denominator is converted to an equivalent fraction whose denominator is an integer. This process is still used today and is useful in other areas of mathematics, too.
Square roots of numbers that are not perfect squares are irrational numbers. When we rationalize the denominator, we write an equivalent fraction with a rational number in the denominator.
Let’s look at a numerical example.
But we can find a fraction equivalent to by multiplying the numerator and denominator by .
Now if we need an approximate value, we divide . This is much easier.
Even though we have calculators available nearly everywhere, a fraction with a radical in the denominator still must be rationalized. It is not considered simplified if the denominator contains a square root.
Similarly, a square root is not considered simplified if the radicand contains a fraction.
To rationalize a denominator, we use the property that . If we square an irrational square root, we get a rational number.
We will use this property to rationalize the denominator in the next example.
Always simplify the radical in the denominator first, before you rationalize it. This way the numbers stay smaller and easier to work with.
### Rationalize a Two-Term Denominator
When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.
When we multiply a binomial that includes a square root by its conjugate, the product has no square roots.
### Key Concepts
1. Quotient Property of Square Roots
2. Simplified Square RootsA square root is considered simplified if there are
### Practice Makes Perfect
Divide Square Roots
In the following exercises, simplify.
Rationalize a One-Term Denominator
In the following exercises, simplify and rationalize the denominator.
Rationalize a Two-Term Denominator
In the following exercises, simplify by rationalizing the denominator.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Roots and Radicals
## Solve Equations with Square Roots
### Solve Radical Equations
In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.
As usual, in solving these equations, what we do to one side of an equation we must do to the other side as well. Since squaring a quantity and taking a square root are ‘opposite’ operations, we will square both sides in order to remove the radical sign and solve for the variable inside.
But remember that when we write we mean the principal square root. So always. When we solve radical equations by squaring both sides we may get an algebraic solution that would make negative. This algebraic solution would not be a solution to the original radical equation; it is an extraneous solution. We saw extraneous solutions when we solved rational equations, too.
Now we will see how to solve a radical equation. Our strategy is based on the relation between taking a square root and squaring.
When we use a radical sign, we mean the principal or positive root. If an equation has a square root equal to a negative number, that equation will have no solution.
If one side of the equation is a binomial, we use the binomial squares formula when we square it.
Don’t forget the middle term!
When there is a coefficient in front of the radical, we must square it, too.
Sometimes after squaring both sides of an equation, we still have a variable inside a radical. When that happens, we repeat Step 1 and Step 2 of our procedure. We isolate the radical and square both sides of the equation again.
### Use Square Roots in Applications
As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.
We will use our Problem Solving Strategy for Geometry Applications, with slight modifications, to give us a plan for solving applications with formulas from any discipline.
We used the formula to find the area of a rectangle with length L and width W. A square is a rectangle in which the length and width are equal. If we let s be the length of a side of a square, the area of the square is .
The formula gives us the area of a square if we know the length of a side. What if we want to find the length of a side for a given area? Then we need to solve the equation for s.
We can use the formula to find the length of a side of a square for a given area.
We will show an example of this in the next example.
Another application of square roots has to do with gravity.
For example, if an object is dropped from a height of 64 feet, we can find the time it takes to reach the ground by substituting into the formula.
It would take 2 seconds for an object dropped from a height of 64 feet to reach the ground.
Police officers investigating car accidents measure the length of the skid marks on the pavement. Then they use square roots to determine the speed, in miles per hour, a car was going before applying the brakes.
### Key Concepts
1. To Solve a Radical Equation:
2. Solving Applications with Formulas
3. Area of a Square
4. Falling Objects
5. Skid Marks and Speed of a Car
### Practice Makes Perfect
Solve Radical Equations
In the following exercises, check whether the given values are solutions.
In the following exercises, solve.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Roots and Radicals
## Higher Roots
### Simplify Expressions with Higher Roots
Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.
Let’s review some vocabulary first.
The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.
It will be helpful to have a table of the powers of the integers from . See .
Notice the signs in . All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of below to help you see this.
Earlier in this chapter we defined the square root of a number.
And we have used the notation to denote the principal square root. So always.
We will now extend the definition to higher roots.
We do not write the index for a square root. Just like we use the word ‘cubed’ for , we use the term ‘cube root’ for .
We refer to to help us find higher roots.
Could we have an even root of a negative number? No. We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.
When we worked with square roots that had variables in the radicand, we restricted the variables to non-negative values. Now we will remove this restriction.
The odd root of a number can be either positive or negative. We have seen that .
But the even root of a non-negative number is always non-negative, because we take the principal .
Suppose we start with .
How can we make sure the fourth root of −5 raised to the fourth power, is 5? We will see in the following property.
### Use the Product Property to Simplify Expressions with Higher Roots
We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An nth root is considered simplified if it has no factors of .
We will generalize the Product Property of Square Roots to include any integer root .
Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.
### Use the Quotient Property to Simplify Expressions with Higher Roots
We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.
Previously, we used the Quotient Property ‘in reverse’ to simplify square roots. Now we will generalize the formula to include higher roots.
If the fraction inside the radical cannot be simplified, we use the first form of the Quotient Property to rewrite the expression as the quotient of two radicals.
### Add and Subtract Higher Roots
We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.
Like radicals have the same index and the same radicand.
1. and are like radicals.
2. and are not like radicals. The radicands are different.
3. and are not like radicals. The indices are different.
We add and subtract like radicals in the same way we add and subtract like terms. We can add and the result is .
When an expression does not appear to have like radicals, we will simplify each radical first. Sometimes this leads to an expression with like radicals.
### Key Concepts
1. Properties of
2. when is an even number and
3. is considered simplified if a has no factors of .
4. Product Property of
5. Quotient Property of
6. To combine like radicals, simply add or subtract the coefficients while keeping the radical the same.
### Practice Makes Perfect
Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Add and Subtract Higher Roots
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
# Roots and Radicals
## Rational Exponents
### Simplify Expressions with
Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.
The Power Property for Exponents says that when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.
Suppose we want to find a number p such that . We will use the Power Property of Exponents to find the value of p.
But we know also . Then it must be that .
This same logic can be used for any positive integer exponent n to show that .
There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you’ll practice converting expressions between these two notations.
In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.
Be careful of the placement of the negative signs in the next example. We will need to use the property in one case.
### Simplify Expressions with
Let’s work with the Power Property for Exponents some more.
Suppose we raise to the power m.
Now suppose we take to the power.
Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller.
Remember that . The negative sign in the exponent does not change the sign of the expression.
### Use the Laws of Exponents to Simplify Expressions with Rational Exponents
The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.
When we multiply the same base, we add the exponents.
We will use the Power Property in the next example.
The Quotient Property tells us that when we divide with the same base, we subtract the exponents.
Sometimes we need to use more than one property. In the next two examples, we will use both the Product to a Power Property and then the Power Property.
We will use both the Product and Quotient Properties in the next example.
### Key Concepts
1. Summary of Exponent Properties
2. If are real numbers and are rational numbers, then
### Section Exercises
### Practice Makes Perfect
Simplify Expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.
### Everyday Math
### Writing Exercises
### Chapter 9 Review Exercises
### Simplify and Use Square Roots
Simplify Expressions with Square Roots
In the following exercises, simplify.
Estimate Square Roots
In the following exercises, estimate each square root between two consecutive whole numbers.
Approximate Square Roots
In the following exercises, approximate each square root and round to two decimal places.
Simplify Variable Expressions with Square Roots
In the following exercises, simplify.
### Simplify Square Roots
Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.
### Add and Subtract Square Roots
Add and Subtract Like Square Roots
In the following exercises, simplify.
Add and Subtract Square Roots that Need Simplification
In the following exercises, simplify.
### Multiply Square Roots
Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
### Divide Square Roots
Divide Square Roots
In the following exercises, simplify.
Rationalize a One Term Denominator
In the following exercises, rationalize the denominator.
Rationalize a Two Term Denominator
In the following exercises, rationalize the denominator.
### Solve Equations with Square Roots
Solve Radical Equations
In the following exercises, solve the equation.
Use Square Roots in Applications
In the following exercises, solve. Round approximations to one decimal place.
### Higher Roots
Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Product Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Expressions with Higher Roots
In the following exercises, simplify.
Add and Subtract Higher Roots
In the following exercises, simplify.
### Rational Exponents
Simplify Expressions with
In the following exercises, write as a radical expression.
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Simplify Expressions with
In the following exercises, write with a rational exponent.
In the following exercises, simplify.
Use the Laws of Exponents to Simplify Expressions with Rational Exponents
In the following exercises, simplify.
### Practice Test
In the following exercises, simplify.
In the following exercises, rationalize the denominator.
In the following exercises, solve.
In the following exercise, solve. |
# Quadratic Equations
## Introduction
The trajectories of fireworks are modeled by quadratic equations. The equations can be used to predict the maximum height of a firework and the number of seconds it will take from launch to explosion. In this chapter, we will study the properties of quadratic equations, solve them, graph them, and see how they are applied as models of various situations. |
# Quadratic Equations
## Solve Quadratic Equations Using the Square Root Property
Quadratic equations are equations of the form , where . They differ from linear equations by including a term with the variable raised to the second power. We use different methods to solve quadratic equations than linear equations, because just adding, subtracting, multiplying, and dividing terms will not isolate the variable.
We have seen that some quadratic equations can be solved by factoring. In this chapter, we will use three other methods to solve quadratic equations.
### Solve Quadratic Equations of the Form ax2 = k Using the Square Root Property
We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation .
We can easily use factoring to find the solutions of similar equations, like and , because 16 and 25 are perfect squares. But what happens when we have an equation like ? Since 7 is not a perfect square, we cannot solve the equation by factoring.
These equations are all of the form .We defined the square root of a number in this way:
This leads to the Square Root Property.
Notice that the Square Root Property gives two solutions to an equation of the form : the principal square root of and its opposite. We could also write the solution as .
Now, we will solve the equation again, this time using the Square Root Property.
What happens when the constant is not a perfect square? Let’s use the Square Root Property to solve the equation .
To use the Square Root Property, the coefficient of the variable term must equal 1. In the next example, we must divide both sides of the equation by 5 before using the Square Root Property.
The Square Root Property started by stating, ‘If , and ’. What will happen if ? This will be the case in the next example.
Remember, we first isolate the quadratic term and then make the coefficient equal to one.
The solutions to some equations may have fractions inside the radicals. When this happens, we must rationalize the denominator.
### Solve Quadratic Equations of the Form a(x − h)2 = k Using the Square Root Property
We can use the Square Root Property to solve an equation like , too. We will treat the whole binomial, , as the quadratic term.
Remember, when we take the square root of a fraction, we can take the square root of the numerator and denominator separately.
We will start the solution to the next example by isolating the binomial.
The left sides of the equations in the next two examples do not seem to be of the form . But they are perfect square trinomials, so we will factor to put them in the form we need.
### Key Concepts
1. Square Root PropertyIf , and , then .
### Practice Makes Perfect
Solve Quadratic Equations of the form Using the Square Root Property
In the following exercises, solve the following quadratic equations.
Solve Quadratic Equations of the Form Using the Square Root Property
In the following exercises, solve the following quadratic equations.
Mixed Practice
In the following exercises, solve using the Square Root Property.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently: Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help: This must be addressed quickly because topics you do not master become potholes in your road to success. In math, every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no-I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need. |
# Quadratic Equations
## Solve Quadratic Equations by Completing the Square
So far, we have solved quadratic equations by factoring and using the Square Root Property. In this section, we will solve quadratic equations by a process called ‘completing the square.’
### Complete The Square of a Binomial Expression
In the last section, we were able to use the Square Root Property to solve the equation because the left side was a perfect square.
We also solved an equation in which the left side was a perfect square trinomial, but we had to rewrite it the form in order to use the square root property.
What happens if the variable is not part of a perfect square? Can we use algebra to make a perfect square?
Let’s study the binomial square pattern we have used many times. We will look at two examples.
We can use this pattern to “make” a perfect square.
We will start with the expression . Since there is a plus sign between the two terms, we will use the pattern.
Notice that the first term of is a square, .
We now know .
What number can we add to to make a perfect square trinomial?
The middle term of the Binomial Squares Pattern, , is twice the product of the two terms of the binomial. This means twice the product of and some number is . So, two times some number must be six. The number we need is The second term in the binomial, must be 3.
We now know .
Now, we just square the second term of the binomial to get the last term of the perfect square trinomial, so we square three to get the last term, nine.
We can now factor to
So, we found that adding nine to ‘completes the square,’ and we write it as .
### Solve Quadratic Equations of the Form x2 + bx + c = 0 by completing the square
In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square, too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.
For example, if we start with the equation and we want to complete the square on the left, we will add nine to both sides of the equation.
Then, we factor on the left and simplify on the right.
Now the equation is in the form to solve using the Square Root Property. Completing the square is a way to transform an equation into the form we need to be able to use the Square Root Property.
In the previous example, there was no real solution because was equal to a negative number.
We will start the next example by isolating the variable terms on the left side of the equation.
To solve the next equation, we must first collect all the variable terms to the left side of the equation. Then, we proceed as we did in the previous examples.
Notice that the left side of the next equation is in factored form. But the right side is not zero, so we cannot use the Zero Product Property. Instead, we multiply the factors and then put the equation into the standard form to solve by completing the square.
### Solve Quadratic Equations of the form ax2 + bx + c = 0 by completing the square
The process of completing the square works best when the leading coefficient is one, so the left side of the equation is of the form . If the term has a coefficient, we take some preliminary steps to make the coefficient equal to one.
Sometimes the coefficient can be factored from all three terms of the trinomial. This will be our strategy in the next example.
To complete the square, the leading coefficient must be one. When the leading coefficient is not a factor of all the terms, we will divide both sides of the equation by the leading coefficient. This will give us a fraction for the second coefficient. We have already seen how to complete the square with fractions in this section.
### Key Concepts
1. Binomial Squares Pattern If are real numbers,
2. Complete a SquareTo complete the square of :
### Practice Makes Perfect
Complete the Square of a Binomial Expression
In the following exercises, complete the square to make a perfect square trinomial. Then, write the result as a binomial squared.
Solve Quadratic Equations of the Form by Completing the Square
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the Form by Completing the Square
In the following exercises, solve by completing the square.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Quadratic Equations
## Solve Quadratic Equations Using the Quadratic Formula
When we solved quadratic equations in the last section by completing the square, we took the same steps every time. By the end of the exercise set, you may have been wondering ‘isn’t there an easier way to do this?’ The answer is ‘yes.’ In this section, we will derive and use a formula to find the solution of a quadratic equation.
We have already seen how to solve a formula for a specific variable ‘in general’ so that we would do the algebraic steps only once and then use the new formula to find the value of the specific variable. Now, we will go through the steps of completing the square in general to solve a quadratic equation for x. It may be helpful to look at one of the examples at the end of the last section where we solved an equation of the form as you read through the algebraic steps below, so you see them with numbers as well as ‘in general.’
This last equation is the Quadratic Formula.
To use the Quadratic Formula, we substitute the values of into the expression on the right side of the formula. Then, we do all the math to simplify the expression. The result gives the solution(s) to the quadratic equation.
If you say the formula as you write it in each problem, you’ll have it memorized in no time. And remember, the Quadratic Formula is an equation. Be sure you start with ‘’.
When we solved quadratic equations by using the Square Root Property, we sometimes got answers that had radicals. That can happen, too, when using the Quadratic Formula. If we get a radical as a solution, the final answer must have the radical in its simplified form.
We cannot take the square root of a negative number. So, when we substitute , , and into the Quadratic Formula, if the quantity inside the radical is negative, the quadratic equation has no real solution. We will see this in the next example.
The quadratic equations we have solved so far in this section were all written in standard form, . Sometimes, we will need to do some algebra to get the equation into standard form before we can use the Quadratic Formula.
When we solved linear equations, if an equation had too many fractions we ‘cleared the fractions’ by multiplying both sides of the equation by the LCD. This gave us an equivalent equation—without fractions—to solve. We can use the same strategy with quadratic equations.
Think about the equation . We know from the Zero Products Principle that this equation has only one solution: .
We will see in the next example how using the Quadratic Formula to solve an equation with a perfect square also gives just one solution.
### Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions. Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation?
Yes, the quantity inside the radical of the Quadratic Formula makes it easy for us to determine the number of solutions. This quantity is called the discriminant.
Let’s look at the discriminant of the equations in , , and , and the number of solutions to those quadratic equations.
When the discriminant is positive the quadratic equation has two solutions.
When the discriminant is zero the quadratic equation has one solution.
When the discriminant is negative the quadratic equation has no real solutions.
### Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
We have used four methods to solve quadratic equations:
1. Factoring
2. Square Root Property
3. Completing the Square
4. Quadratic Formula
You can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method to use.
What about the method of completing the square? Most people find that method cumbersome and prefer not to use it. We needed to include it in this chapter because we completed the square in general to derive the Quadratic Formula. You will also use the process of completing the square in other areas of algebra.
### Key Concepts
1. Quadratic Formula The solutions to a quadratic equation of the form are given by the formula:
2. Solve a Quadratic Equation Using the Quadratic Formula
To solve a quadratic equation using the Quadratic Formula.
3. Using the Discriminant,
For a quadratic equation of the form
4. To identify the most appropriate method to solve a quadratic equation:
### Practice Makes Perfect
Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation
In the following exercises, determine the number of solutions to each quadratic equation.
Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation. Do not solve.
### Everyday Math
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve? |
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