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# Quadratic Equations
## Solve Applications Modeled by Quadratic Equations
### Solve Applications of the Quadratic Formula
We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even integers and consecutive odd integers by modeling the situation with linear equations. Remember, we noticed each even 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 . This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
Some applications of consecutive odd integers or consecutive even integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
We will use the formula for the area of a triangle to solve the next example.
Recall that, when we solve geometry applications, it is helpful to draw the figure.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
The height of a projectile shot upwards is modeled by a quadratic equation. The initial velocity, , propels the object up until gravity causes the object to fall back down.
We can use the formula for projectile motion to find how many seconds it will take for a firework to reach a specific height.
### Key Concepts
1. Area of a Triangle For a triangle with base, , and height, , the area, , is given by the formula:
2. Pythagorean Theorem In any right triangle, where and are the lengths of the legs, and is the length of the hypothenuse,
3. Projectile motion The height in feet, , of an object shot upwards into the air with initial velocity, , after seconds can be modeled by the formula:
### Practice Makes Perfect
Solve Applications of the Quadratic Formula
In the following exercises, solve by using methods of factoring, the square root principle, or the Quadratic Formula. Round your answers 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.
ⓑ 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? |
# Quadratic Equations
## Graphing Quadratic Equations in Two Variables
### Recognize the Graph of a Quadratic Equation in Two Variables
We have graphed equations of the form . We called equations like this linear equations because their graphs are straight lines.
Now, we will graph equations of the form . We call this kind of equation a quadratic equation in two variables.
Just like we started graphing linear equations by plotting points, we will do the same for quadratic equations.
Let’s look first at graphing the quadratic equation . We will choose integer values of between and 2 and find their values. See .
Notice when we let and , we got the same value for .
The same thing happened when we let and .
Now, we will plot the points to show the graph of . See .
The graph is not a line. This figure is called a parabola. Every quadratic equation has a graph that looks like this.
In you will practice graphing a parabola by plotting a few points.
How do the equations and differ? What is the difference between their graphs? How are their graphs the same?
All parabolas of the form open upwards or downwards. See .
Notice that the only difference in the two equations is the negative sign before the in the equation of the second graph in . When the term is positive, the parabola opens upward, and when the term is negative, the parabola opens downward.
### Find the Axis of Symmetry and Vertex of a Parabola
Look again at . Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry in blue. See .
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of is
So, to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula .
Look back at . Are these the equations of the dashed red lines?
The point on the parabola that is on the axis of symmetry is the lowest or highest point on the parabola, depending on whether the parabola opens upwards or downwards. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means its x-coordinate is . To find the y-coordinate of the vertex, we substitute the value of the x-coordinate into the quadratic equation.
### Find the Intercepts of a Parabola
When we graphed linear equations, we often used the x- and y-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the the value of is zero. So, to find the y-intercept, we substitute into the equation.
Let’s find the y-intercepts of the two parabolas shown in the figure below.
At an , the value of is zero. To find an x-intercept, we substitute into the equation. In other words, we will need to solve the equation for .
But solving quadratic equations like this is exactly what we have done earlier in this chapter.
We can now find the x-intercepts of the two parabolas shown in .
First, we will find the with equation .
Now, we will find the x-intercepts of the parabola with equation .
We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph.
Do these results agree with our graphs? See .
In this chapter, we have been solving quadratic equations of the form . We solved for and the results were the solutions to the equation.
We are now looking at quadratic equations in two variables of the form . The graphs of these equations are parabolas. The x-intercepts of the parabolas occur where .
For example:
The solutions of the quadratic equation are the values of the x-intercepts.
Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the equations give the x-intercepts of the graphs, the number of x-intercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic equation of the form . Now, we can use the discriminant to tell us how many x-intercepts there are on the graph.
Before you start solving the quadratic equation to find the values of the x-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.
### Graph Quadratic Equations in Two Variables
Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.
We were able to find the x-intercepts in the last example by factoring. We find the x-intercepts in the next example by factoring, too.
For the graph of , the vertex and the x-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation is 0, so there is only one solution. That means there is only one x-intercept, and it is the vertex of the parabola.
How many x-intercepts would you expect to see on the graph of ?
Finding the y-intercept by substituting into the equation is easy, isn’t it? But we needed to use the Quadratic Formula to find the x-intercepts in . We will use the Quadratic Formula again in the next example.
### Solve Maximum and Minimum Applications
Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The y-coordinate of the vertex is the minimum y-value of a parabola that opens upward. It is the maximum y-value of a parabola that opens downward. See .
We have used the formula
to calculate the height in feet, , of an object shot upwards into the air with initial velocity, , after seconds.
This formula is a quadratic equation in the variable , so its graph is a parabola. By solving for the coordinates of the vertex, we can find how long it will take the object to reach its maximum height. Then, we can calculate the maximum height.
### Key Concepts
1. The graph of every quadratic equation is a parabola.
2. Parabola Orientation For the quadratic equation , if
3. Axis of Symmetry and Vertex of a Parabola For a parabola with equation :
4. Find the Intercepts of a Parabola To find the intercepts of a parabola with equation :
5. To Graph a Quadratic Equation in Two Variables
6. Minimum or Maximum Values of a Quadratic Equation
### Section Exercises
### Practice Makes Perfect
Recognize the Graph of a Quadratic Equation in Two Variables
In the following exercises, graph:
In the following exercises, determine if the parabola opens up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
Find the Intercepts of a Parabola
In the following exercises, find the x- and y-intercepts.
Graph Quadratic Equations in Two Variables
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value.
In the following exercises, solve. Round answers 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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
### Chapter 10 Review Exercises
### 10.1 Solve Quadratic Equations Using the Square Root Property
In the following exercises, solve using the Square Root Property.
In the following exercises, solve using the Square Root Property.
### 10.2 Solve Quadratic Equations Using Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.
### 10.3 Solve Quadratic Equations Using the Quadratic Formula
In the following exercises, solve by using the Quadratic Formula.
In the following exercises, determine the number of solutions to each quadratic equation.
In the following exercises, identify the most appropriate method (Factoring, Square Root, or Quadratic Formula) to use to solve each quadratic equation.
### 10.4 Solve Applications Modeled by Quadratic Equations
In the following exercises, solve by using methods of factoring, the square root principle, or the quadratic formula.
### 10.5 Graphing Quadratic Equations in Two Variables
In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
In the following exercises, find ⓐ the axis of symmetry and ⓑ the vertex.
In the following exercises, find the x- and y-intercepts.
In the following exercises, graph by using intercepts, the vertex, and the axis of symmetry.
In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.
### Practice Test
Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number of solutions of each quadratic equation.
Solve by factoring, the Square Root Property, or the Quadratic Formula.
For each parabola, find ⓐ which ways it opens, ⓑ the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and ⓔ the maximum or minimum value.
Graph the following parabolas by using intercepts, the vertex, and the axis of symmetry.
Solve. |
# Foundations
## Introduction
For years, doctors and engineers have worked to make artificial limbs, such as this hand for people who need them. This particular product is different, however, because it was developed using a 3D printer. As a result, it can be printed much like you print words on a sheet of paper. This makes producing the limb less expensive and faster than conventional methods.
Biomedical engineers are working to develop organs that may one day save lives. Scientists at NASA are designing ways to use 3D printers to build on the moon or Mars. Already, animals are benefitting from 3D-printed parts, including a tortoise shell and a dog leg. Builders have even constructed entire buildings using a 3D printer.
The technology and use of 3D printers depend on the ability to understand the language of algebra. Engineers must be able to translate observations and needs in the natural world to complex mathematical commands that can provide directions to a printer. In this chapter, you will review the language of algebra and take your first steps toward working with algebraic concepts. |
# Foundations
## Use the Language of Algebra
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.
### Find Factors, Prime Factorizations, and Least Common Multiples
The numbers 2, 4, 6, 8, 10, 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.
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
If we were to look for patterns in the multiples of the numbers 2 through 9, we would discover the following divisibility tests:
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 and The number 72 has many factors: and
Some numbers, such as 72, have many factors. Other numbers have only two factors. A prime number is a counting number greater than 1 whose only factors are 1 and itself.
The counting numbers from 2 to 20 are listed in the table 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 in many topics 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. Otherwise it is easy to lose track of the prime numbers.
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 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.
To find the least common multiple of two numbers we will use the Prime Factors Method. Let’s find the LCM of 12 and 18 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.
### Use Variables and Algebraic Symbols
In algebra, we use a letter of the alphabet to represent a number whose value may change. We call this a variable and letters commonly used for variables are
A number whose value always remains the same is called a constant.
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 below.
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 or can be read from left to right or right to left, though in English we usually read from left to right. In general,
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in English. They help identify an expression, which can be made up of number, a variable, or a combination of numbers and variables using operation symbols. We will introduce three types of grouping symbols 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. A sentence has a subject and a verb. In algebra, we have expressions and equations.
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 by 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.
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.
While we read as to the power”, we usually read:
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 would 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 Some students simplify this getting 49, by adding and then multiplying that result by 7. Others get 25, by multiplying first and then adding 4.
The same expression should give the same result. So mathematicians 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.
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 and
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 is 3. When we write the coefficient is 1, since
Some terms share common traits. When two terms are constants or have the same variable and exponent, we say they are like terms.
Look at the following 6 terms. Which ones seem to have traits in common?
We say,
and are like terms.
and are like terms.
and are like terms.
If there are like terms in an expression, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.
### Translate an English Phrase to an Algebraic Expression
We listed many operation symbols that are used in algebra. Now, we will use them to 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.
We look carefully at the words to help us distinguish between multiplying a sum and adding a product.
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.
The expressions in the next example will be used in the typical coin mixture problems we will see soon.
### Key Concepts
1. Divisibility Tests
A number is divisible by:
2 if the last digit is 0, 2, 4, 6, or 8.
3 if the sum of the digits is divisible by 3.
5 if the last digit is 5 or 0.
6 if it is divisible by both 2 and 3.
10 if it ends with 0.
2. How to find the prime factorization of a composite number.
3. How To Find the least common multiple using the prime factors method.
4. Equality Symbol
is read “a is equal to b.”
The symbol “=” is called the equal sign.
5. Inequality
6. Inequality Symbols
7. Grouping Symbols
8. Exponential Notation
means multiply a by itself, n times.
The expression is read a to the power.
9. Simplify an Expression
To simplify an expression, do all operations in the expression.
10. How to use the order of operations.
11. How to combine like terms.
### Practice Makes Perfect
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 each pair of numbers using the prime factors method.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
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.
### Writing Exercises
### Self Check
ⓐ 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
## Integers
### Simplify Expressions with Absolute Value
A negative number is a number less than 0. The negative numbers are to the left of zero on the number line. See .
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 and are the same distance from zero, each one is called the opposite of the other. The opposite of is and the opposite of is
illustrates the definition.
We saw that numbers such as 3 and are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.
For example,
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 0 to 0 on the number line is zero units.
In the next example, we’ll order expressions with absolute values.
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 and Subtract Integers
So far, we have only used the counting numbers and the whole numbers.
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
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 first example,
adds 5 positives and 3 positives—both positives.
The second example, adds 5 negatives and 3 negatives—both negatives.
When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.
So what happens when the signs are different? Let’s add and
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.
We will continue to use counters to model the subtraction. Perhaps when you were younger, you read as “5 take away 3.” When you use counters, you can think of subtraction the same way!
We will use the counters to show to subtract:
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. If we don’t have the number of counters needed to take away, we add 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.
Let’s look at and
Have you noticed that subtraction of signed numbers can be done by adding the opposite? In the last example, is the same as and is the same as You will often see this idea, the Subtraction Property, written as follows:
What happens when there are more than three integers? We just use the order of operations as usual.
### Multiply and Divide 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 are using 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 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.
In summary:
Notice that for multiplication of two signed numbers, when the
What about division? Division is the inverse operation of multiplication. So, because In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.
Division follows the same rules as multiplication with regard to signs.
When we multiply a number by 1, the result is the same number. Each time we multiply a number by we get its opposite!
### 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 Please Excuse 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 last example showed us the difference between and This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.
### 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.
### 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. Opposite Notation
2. Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as and for all numbers.
Absolute values are always greater than or equal to zero.
3. Grouping Symbols
4. Subtraction Property
Subtracting a number is the same as adding its opposite.
5. Multiplication and Division of Signed Numbers
For multiplication and division of two signed numbers:
If the signs are the same, the result is positive.
If the signs are different, the result is negative.
6. Multiplication by
Multiplying a number by gives its opposite.
7. How to Use Integers in Applications.
### Practice Makes Perfect
Simplify Expressions with Absolute Value
In the following exercises, fill in or for each of the following pairs of numbers.
In the following exercises, simplify.
Add and Subtract Integers
In the following exercises, simplify each expression.
Multiply and Divide Integers
In the following exercises, multiply or divide.
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
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.
### 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
## Fractions
### Simplify Fractions
A fraction is a way to represent parts of a whole. The fraction represents two of three equal parts. See . In the fraction the 2 is called the numerator and the 3 is called the denominator. The line is called the fraction bar.
Fractions that have the same value are equivalent fractions. The Equivalent Fractions
Property allows us to find equivalent fractions and also simplify fractions.
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator.
For example,
is simplified because there are no common factors of 2 and
is not simplified because 5 is a common factor of 10 and
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.
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 and Divide Fractions
Many people find multiplying and dividing fractions easier than adding and subtracting fractions.
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 a negative by a negative, so the product will be positive.
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,
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 Since 4 is written in fraction form as the reciprocal of 4 is
To divide fractions, we multiply the first fraction by the reciprocal of the second.
We need to say and to be sure we don’t divide by zero!
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, remember that the fraction bar means division. For example, the complex fraction means
### Add and Subtract Fractions
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.
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!
We now have all four operations for fractions. summarizes fraction operations.
When starting an exercise, always identify the operation and then recall the methods needed for that operation.
### Use the Order of Operations to Simplify Fractions
The fraction bar in a fraction acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. 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.
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 as the fraction bar means division.
### 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.
### Key Concepts
1. Equivalent Fractions Property
If a, b, and c are numbers where then
2. How to simplify a fraction.
3. Fraction Multiplication
If a, b, c, and d are numbers where and then
To multiply fractions, multiply the numerators and multiply the denominators.
4. Fraction Division
If a, b, c, and d are numbers where and then
To divide fractions, we multiply the first fraction by the reciprocal of the second.
5. Fraction Addition and Subtraction
If a, b, and c are numbers where then
To add or subtract fractions, add or subtract the numerators and place the result over the common denominator.
6. How to add or subtract fractions.
7. How to simplify an expression with a fraction bar.
8. Placement of Negative Sign in a Fraction
For any positive numbers a and b,
9. How to simplify complex fractions.
### Practice Makes Perfect
Simplify Fractions
In the following exercises, simplify.
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
In the following exercises, simplify.
Add and Subtract Fractions
In the following exercises, add or subtract.
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
Mixed Practice
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
### 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
## Decimals
### Round Decimals
Decimals are another way of writing fractions whose denominators are powers of ten.
Just as in whole numbers, each digit of a decimal corresponds to the place value based on the powers of ten. shows the names of the place values to the left and right of the decimal point.
When we work with decimals, it is often necessary to round the number to the nearest required place value. 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
When we multiply signed decimals, first we determine the sign of the product and then multiply as if the numbers were both positive. We multiply the numbers temporarily ignoring the decimal point and then count the number of decimal points in the factors and that sum tells us the number of decimal places in the product. Finally, we write the product with the appropriate sign.
Often, 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 signed decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed and the sign of the quotient. 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.
### Convert Decimals, Fractions, and Percents
In our work, it is often necessary to change the form of a number. We may have to change fractions to decimals or decimals to percent.
We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.
The steps to take to convert a decimal to a fraction are summarized in the procedure box.
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. After doing this many times, you may see the pattern.
To convert a percent number to a decimal number, we move the decimal point two places to the left.
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. After many conversions, you may 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.
### Simplify Expressions with Square Roots
Remember that when a number is multiplied by itself, we write and read it “ squared.” The result is called the square of a number n. For example, is read “8 squared” and 64 is called the square of 8. Similarly, 121 is the square of 11 because is 121. It will be helpful to learn to recognize the perfect square 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.
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 m is called a square root of a number 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. 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 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 principal square root of 10.”
### Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
We have already described numbers as counting numbers, whole numbers, and integers. What is the difference between these types of numbers? Difference could be confused with subtraction. How about asking how we distinguish 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.
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. The decimal for is the number The bar over the 3 indicates that the number 3 repeats infinitely. Continuously has an important meaning in calculus. The number(s) under the bar is called the repeating block and it repeats continuously.
Since all integers can be written as a fraction whose denominator is 1, the integers (and so also the counting and whole numbers. are rational numbers.
Every rational number can be written both as a ratio of integers where p and q are integers and and as a decimal that stops or repeats.
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 use three dots (…) to indicate the decimal does not stop or repeat.
The square root of a number that is not a perfect square is a decimal that does not stop or repeat.
A numbers whose decimal form does not stop or repeat cannot be written as a fraction of integers. We call this an irrational number.
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.
Later in this course we will introduce numbers beyond the real numbers. illustrates how the number sets we’ve used so far fit together.
Does the term “real numbers” seem strange to you? Are there any numbers that are not “real,” and, if so, what could they be? 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 and Decimals on the Number Line
We now want to include fractions and decimals on the number line. Let’s start with fractions and locate and on the number line.
We’ll start with the whole numbers 3 and because they are the easiest to plot. See .
The proper fractions listed are and We know the proper fraction has value less than one and so would be located between 0 and 1. The denominator is 5, so we divide the unit from 0 to 1 into 5 equal parts We plot
Similarly, is between 0 and After dividing the unit into 5 equal parts we plot
Finally, look at the improper fractions Locating these points may be easier if you change each of them to a mixed number.
shows the number line with all the points plotted.
Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.
### Key Concepts
1. How to round decimals.
2. How to add or subtract decimals.
3. How to multiply decimals.
4. How to multiply a decimal by a power of ten.
5. How to divide decimals.
6. How to convert a decimal to a proper fraction and a fraction to a decimal.
7. How to convert a percent to a decimal and a decimal to a percent.
8. Square Root Notation
is read “the square root of m.”
If then for
The square root of m, is the positive number whose square is m.
9. Rational or Irrational
If the decimal form of a number
10. Real Numbers
### Practice Makes Perfect
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. Round money monetary answers to the nearest cent.
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.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.
### Writing Exercises
### Self Check
ⓐ 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
## Properties of Real Numbers
### Use the Commutative and Associative Properties
The order we add two numbers doesn’t affect the result. If we add or the results are the same—they both equal 17. So, The order in which we add does not matter!
Similarly, when multiplying two numbers, the order does not affect the result. If we multiply or the results are the same—they both equal 72. So, The order in which we multiply does not matter!
These examples illustrate the Commutative Property.
The Commutative Property has to do with order. We subtract and , and see that Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.
Division is not commutative either. Since changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!
Addition and multiplication are commutative.
Subtraction and division are not commutative.
When adding three numbers, changing the grouping of the numbers gives the same result. For example, since each side of the equation equals 17.
This is true for multiplication, too. For example, since each side of the equation equals 5.
These examples illustrate the Associative Property.
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 Property or Associative Property first.
### Use the Properties of Identity, Inverse, and Zero
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. 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. The Identity Property of Multiplication that states that for any real number and
We summarize the Identity Properties here.
What number added to 5 gives the additive identity, 0? We know
The missing number was the opposite of the number!
We call the additive inverse of 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
What number multiplied by gives the multiplicative identity, 1? In other words, times what results in 1? We know
The missing number was the reciprocal of the number!
We call the multiplicative inverse of a. The reciprocal of a number is its multiplicative inverse. This leads to the Inverse Property of Multiplication that states that for any real number
We’ll formally state the inverse properties here.
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 Think about the related multiplication fact:
Is there a number that multiplied by 0 gives 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 here.
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.
The next example makes us aware of the distinction between dividing 0 by some number or some number being divided by 0.
### 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.
In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.
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 the next example 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!
In the next example, we 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 here.
### Key Concepts
### Section Exercises
### Practice Makes Perfect
Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.
### 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 Review Exercises
### Use the Language of Algebra
Identify Multiples and Factors
In the following exercises, simplify each expression.
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Simplify Expressions by Combining Like Terms
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.
### Integers
Simplify Expressions with Absolute Value
In the following exercise, fill in or for each of the following pairs of numbers.
In the following exercises, simplify.
Add and Subtract Integers
In the following exercises, simplify each expression.
Multiply and Divide Integers
In the following exercise, multiply or divide.
Simplify and Evaluate Expressions with Integers
In the following exercises, simplify each expression.
For 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 exercise, solve.
### Fractions
Simplify Fractions
In the following exercises, simplify.
Multiply and Divide Fractions
In the following exercises, perform the indicated operation.
Add and Subtract Fractions
In the following exercises, perform the indicated operation.
Use the Order of Operations to Simplify Fractions
In the following exercises, simplify.
Evaluate Variable Expressions with Fractions
In the following exercises, evaluate.
### Decimals
Round Decimals
Add and Subtract Decimals
In the following exercises, perform the indicated operation.
Multiply and Divide Decimals
In the following exercises, perform the indicated operation.
Convert Decimals, Fractions and Percents
In the following exercises, convert each decimal to a fraction.
In the following exercises, convert each fraction to a decimal.
In the following exercises, convert each decimal to a percent.
Simplify Expressions with Square Roots
In the following exercises, simplify.
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercise, list the ⓐ whole numbers ⓑ integers ⓒ rational numbers ⓓ irrational numbers ⓔ real numbers for each set of numbers
Locate Fractions and Decimals on the Number Line
In the following exercises, locate the numbers on a number line.
### Properties of Real Numbers
Use the Commutative and Associative Properties
In the following exercises, simplify.
Use the Properties of Identity, Inverse and Zero
In the following exercises, simplify.
Simplify Expressions Using the Distributive Property
In the following exercises, simplify using the Distributive Property.
### Practice Test
In the following exercises, simplify each expression. |
# Solving Linear Equations
## Introduction
Imagine being a pilot, but not just any pilot—a drone pilot. Drones, or unmanned aerial vehicles, are devices that can be flown remotely. They contain sensors that can relay information to a command center where the pilot is located. Larger drones can also carry cargo. In the near future, several companies hope to use drones to deliver materials and piloting a drone will become an important career. Law enforcement and the military are using drones rather than send personnel into dangerous situations. Building and piloting a drone requires the ability to program a set of actions, including taking off, turning, and landing. This, in turn, requires the use of linear equations. In this chapter, you will explore linear equations, develop a strategy for solving them, and relate them to real-world situations. |
# Solving Linear Equations
## Use a General Strategy to Solve Linear Equations
### Solve Linear Equations Using a General Strategy
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 makes it 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!
To determine whether a number is a solution to an equation, we substitute the value for the variable in the equation. If the resulting equation is a true statement, then the number is a solution of the equation.
There are many types of equations that we will learn to solve. In this section we will focus on a linear equation.
To solve a linear equation it is a good idea to have an overall strategy that can be used to solve any linear equation. In the next example, we will give the steps of a general strategy for solving any linear equation. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.
These steps are summarized in the General Strategy for Solving Linear Equations below.
We can solve equations by getting all the variable terms to either side of the equal sign. By collecting the variable terms on the side where the coefficient of the variable is larger, we avoid working with some negatives. This will be a good strategy when we solve inequalities later in this chapter. It also helps us prevent errors with negatives.
### Classify Equations
Whether or not an equation is true depends on the value of the variable. The equation is true when we replace the variable, x, with the value but not true when we replace x with any other value. An equation like this is called a conditional equation. 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.
Solve:
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 is called an identity.
What happens when we solve the equation
Solve:
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.
The next few examples will ask us to classify an equation as conditional, an identity, or as a contradiction.
We summarize the methods for classifying equations in the table.
### Solve Equations with Fraction or Decimal Coefficients
We could use the General Strategy to solve the next example. This method would work 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 (LCD) 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.
To clear an equation of decimals, we think of all the decimals in their fraction form and then find the LCD of those denominators.
Notice in the previous example, 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’ll distribute before we clear the fractions.
When you multiply both sides of an equation by the LCD of the fractions, make sure you multiply each term by the LCD—even if it does not contain a fraction.
Some equations have decimals in them. This kind of equation may 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 ones we will see in the money applications in a later section. Notice that we will clear all decimals by multiplying by the LCD of their fraction form.
### Key Concepts
1. How to determine whether a number is a solution to an equation
2. How to Solve Linear Equations Using a General Strategy
3. How to Solve Equations with Fraction or Decimal Coefficients
### Practice Makes Perfect
Solve Equations Using the General Strategy
In the following exercises, determine whether the given values are solutions to the equation.
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 Fraction or Decimal Coefficients
In the following exercises, solve each equation with fraction 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.
ⓑ 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. |
# Solving Linear Equations
## Use a Problem Solving Strategy
Have you ever had any negative experiences in the past with word problems? When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. Realize that your negative experiences with word problems are in your past. To move forward you need to calm your fears and change your negative feelings.
Start with a fresh slate and begin to think positive thoughts. Repeating some of the following statements may be helpful to turn your thoughts positive. Thinking positive thoughts is a first step towards success.
I think I can! I think I can!
While word problems were hard in the past, I think I can try them now.
I am better prepared now—I think I will begin to understand word problems.
I am able to solve equations because I practiced many problems and I got help when I needed it—I can try that with word problems.
It may take time, but I can begin to solve word problems.
You are now well prepared and you are ready to succeed. If you take control and believe you can be successful, you will be able to master word problems.
### Use a Problem Solving Strategy for Word Problems
Now that we can solve equations, we are ready to apply our new skills to word problems. We will develop a strategy we can use to solve any word problem successfully.
We summarize an effective strategy for problem solving.
### Solve Number Word Problems
We will now apply the problem solving strategy to “number word problems.” Number word problems give some clues about one or more numbers and we use these clues to write an equation. Number word problems provide good practice for using the Problem Solving Strategy.
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. Therefore, 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
We will use this notation to represent consecutive integers in the next example.
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 two more than the number preceding it. If we call the first one n, then the next one is The one after that would be or
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 63, 65, and 67.
Does it seem strange to have to add two (an even number) to get the next odd number? Do we 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 do not have to do anything different. The pattern is still the same—to get to the next odd or the next even integer, add two.
When a number problem is in a real life context, we still use the same strategies that we used for the previous examples.
### Solve Percent Applications
There are several methods to solve percent equations. In algebra, it is easiest if we just translate English sentences into algebraic equations and then solve the equations. Be sure to change the given percent to a decimal before you use it in the equation.
Now that we have a problem solving 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 we will solve involve everyday situations, you can rely on your own experience.
Remember to put the answer in the form requested. In the next example we are looking for the percent.
It is often important in many fields—business, sciences, pop culture—to talk about how much an amount has increased or decreased over a certain period of time. This increase or decrease is generally expressed as a percent and called the percent change.
To find the percent change, first we find the amount of change, by finding the difference of the new amount and the original amount. Then we find what percent the amount of change is of the original amount.
Applications of discount and mark-up 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.
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.
### Solve Simple Interest Applications
Interest is a part of our daily lives. From the interest earned on our savings to the interest we pay on a car loan or credit card debt, we all have some experience with interest in our lives.
The amount of money you initially deposit into a bank is called the principal, P, and the bank pays you interest, I. When you take out a loan, you pay interest on the amount you borrow, also called the principal.
In either case, the interest is computed as a certain percent of the principal, called the rate of interest, r. The rate of interest is usually expressed as a percent per year, and is calculated by using the decimal equivalent of the percent. The variable t, (for time) represents the number of years the money is saved or borrowed.
Interest is calculated as simple interest or compound interest. Here we will use simple interest.
The formula we use to calculate interest is To use the formula we substitute in the values for variables that are given, 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 do not know the rate.
In the next example, we are asked to find the principal—the amount borrowed.
### Key Concepts
1. How To Use a Problem Solving Strategy for Word Problems
2. How To Find Percent Change
3. Discount
4. Mark-up
5. Simple Interest
If an amount of money, P, called the principal, is invested or borrowed for a period of t years at an annual interest rate r, the amount of interest, I, earned or paid is:
### Practice Makes Perfect
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 Word Problems
In the following exercises, solve each number word problem.
Solve Percent Applications
In the following exercises, translate and solve.
In the following exercises, solve.
In the following exercises, solve.
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.
Solve Simple Interest 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 objective of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives? |
# Solving Linear Equations
## Solve a Formula for a Specific Variable
### Solve a Formula for a Specific Variable
We have all probably worked with some geometric formulas in our study of mathematics. Formulas are used in so many fields, it is important to recognize formulas and be able to manipulate them easily.
It is often helpful to solve a formula for a specific variable. If you need to put a formula in a spreadsheet, it is not unusual to have to solve it for a specific variable first. We isolate that variable on one side of the equals sign with a coefficient of one and all other variables and constants are on the other side of the equal sign.
Geometric formulas often need to be solved for another variable, too. The formula is used to find the volume of a right circular cone when given the radius of the base and height. In the next example, we will solve this formula for the height.
In the sciences, we often need to change temperature from Fahrenheit to Celsius or vice versa. If you travel in a foreign country, you may want to change the Celsius temperature to the more familiar Fahrenheit temperature.
The next example uses the formula for the surface area of a right cylinder.
Sometimes we 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.
### Use Formulas to Solve Geometry Applications
In this objective 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.
When we solve geometry applications, we often have to use some of the properties of the figures. We will review those properties as needed.
The next example involves the area of a triangle. The area of a triangle is one-half the base times the height. We can write this as where b = length of the base and h = height.
In the next example, we will work with a right triangle. To solve for the measure of each angle, we need to use two triangle properties. In any triangle, the sum of the measures of the angles is We can write this as a formula: Also, since the triangle is a right triangle, we remember that a right triangle has one angle.
Here, 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.
The next example uses another important geometry formula. The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. Writing the formula in every exercise and saying it aloud as you write it may help you memorize the Pythagorean Theorem.
We will use the Pythagorean Theorem in the next example.
The next example is about the perimeter of a rectangle. Since the perimeter is just the distance around the rectangle, we find the sum of the lengths of its four sides—the sum of two lengths and two widths. We can write is as where L is the length and is the width. To solve the example, we will need to define the length in terms of the width.
The next example is about the perimeter of a triangle. Since the perimeter is just the distance around the triangle, we find the sum of the lengths of its three sides. We can write this as where a, b, and c are the lengths of the sides.
Applications of these geometric properties can be found in many everyday situations as shown in the next example.
### Key Concepts
1. How To Solve Geometry Applications
2. The Pythagorean Theorem
### Practice Makes Perfect
Solve a Formula for a Specific Variable
In the following exercises, solve the given formula for the specified variable.
In the following exercises, solve for the formula for y.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
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 a geometry formula.
In the following exercises, solve. Approximate answers to the nearest tenth, if necessary.
### 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
## Solve Mixture and Uniform Motion Applications
### Solve Coin Word Problems
Using algebra to find the number of nickels and pennies in a piggy bank may seem silly. You may wonder why we just don’t open the bank and count them. But this type of problem introduces us to some techniques that will be useful as we move forward in our study of mathematics.
If we have a pile of dimes, how would we determine its value? 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 23 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 23 dimes, multiply 23 by $0.10 to get $2.30.
The number of dimes times the value of each dime equals the total value of the dimes.
This method leads to the following model.
If we had several types of coins, 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.
The steps for solving a coin word problem are summarized below.
### 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.
In most of our examples so far, we have been told that one quantity is four more than twice the other, or something similar. In our next example, we have to relate the quantities in a different way.
Suppose Aniket sold a total of 100 tickets. Each ticket was either an adult ticket or a child ticket. If he sold 20 child tickets, how many adult tickets did he sell?
Did you say “80”? How did you figure that out? Did you subtract 20 from 100?
If he sold 45 child tickets, how many adult tickets did he sell?
Did you say “55”? How did you find it? By subtracting 45 from 100?
Now, suppose Aniket sold x child tickets. Then how many adult tickets did he 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 in the table.
We will apply this technique in the next example.
### Solve Mixture Word Problems
Now we’ll solve some more general applications of the mixture model. In mixture problems, we are often mixing two quantities, such as raisins and nuts, to create a mixture, such as trail mix. In our tables we will have a row for each item to be mixed as well as one for the final mixture.
### Solve Uniform Motion Applications
When you are driving down the interstate using your cruise control, the speed of your car stays the same—it is uniform. We call a problem in which the speed of an object is constant a uniform motion application. We will use the distance, rate, and time formula, to compare two scenarios, such as two vehicles travelling at different rates or in opposite directions.
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 coin, ticket, and stamp applications.
The steps are listed here for easy reference:
In , we had two bikers traveling the same distance. In the next example, two people drive toward each other until they meet.
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?
It is important to make sure that the units match when we use the distance rate and time formula. For instance, if the rate is in miles per hour, then the time must be in hours.
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. Total Value of CoinsFor the same type of coin, the total value of a number of coins is found by using the model
2. How to solve coin word problems.
3. How To Solve a Uniform Motion Application
### 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.
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.
ⓑ 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
## Solve Linear Inequalities
### Graph Inequalities on the Number Line
What number would make the inequality true? Are you thinking, “x could be four”? That’s correct, but x could be 6, too, or 37, or even 3.001. Any number greater than three is a solution to the inequality
We show all the solutions to the inequality on the number line by shading in all the numbers to the right of three, to show that all numbers greater than three are solutions. Because the number three itself is not a solution, we put an open parenthesis at three.
We can also represent inequalities using interval notation. 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.
We use the left parenthesis symbol, (, to show that the endpoint of the inequality is not included. The left bracket symbol, [, shows that the endpoint is included.
The inequality means all numbers less than or equal to one. Here we need to show that one is a solution, too. We do that by putting a bracket at We then shade in all the numbers to the left of one, to show that all numbers less than one are solutions. See .
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.
The notation for inequalities on a number line and in interval notation use the same symbols to express the endpoints of intervals.
What numbers are greater than two but less than five? Are you thinking say, ? We can represent all the numbers between two and five with the inequality We can show on the number line by shading all the numbers between two and five. Again, we use the parentheses to show the numbers two and five are not included. See .
### Solve Linear Inequalities
A linear inequality is much like a linear equation—but the equal sign is replaced with an inequality sign. A linear inequality is an inequality in one variable that can be written in one of the forms, or
When we solved linear equations, we were able to use the properties of equality to add, subtract, multiply, or divide both sides and still keep the equality. Similar properties hold true for inequalities.
We can add or subtract the same quantity from both sides of an inequality and still keep the inequality. For example:
Notice that the inequality sign stayed the same.
This leads us to the Addition and Subtraction Properties of Inequality.
What happens to an inequality when we divide or multiply both sides by a constant?
Let’s first multiply and divide both sides by a positive number.
The inequality signs stayed the same.
Does the inequality stay the same when we divide or multiply by a negative number?
Notice that when we filled in the inequality signs, the inequality signs reversed their direction.
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.
This gives us the Multiplication and Division Property of Inequality.
When we divide or multiply an inequality by a:
1. positive number, the inequality stays the same.
2. negative number, the inequality reverses.
Sometimes when solving an inequality, as in the next example, the variable ends upon the right. We can rewrite the inequality in reverse to get the variable to the left.
Think about it as “If Xander is taller than Andy, then Andy is shorter than Xander.”
Be careful when you multiply or divide by a negative number—remember to reverse the inequality sign.
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 make sure to pay close attention when we multiply or divide to isolate the variable.
When solving inequalities, it is usually easiest to collect the variables on the side where the coefficient of the variable is largest. This eliminates negative coefficients and so we don’t have to multiply or divide by a negative—which means we don’t have to remember to reverse the inequality sign.
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.
We can clear fractions in inequalities much as we did in equations. Again, be careful with the signs when multiplying or dividing by a negative.
### 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. shows some common phrases that indicate inequalities.
### Solve Applications with Linear Inequalities
Many real-life situations require us to solve inequalities. 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.
Profit is the money that remains when the costs have been subtracted from the revenue. In the next example, we will find the number of jobs a small businesswoman needs to do every month in order to make a certain amount of profit.
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. Inequalities, Number Lines, and Interval Notation
2. Linear Inequality
3. Addition and Subtraction Property of Inequality
4. Multiplication and Division Property of Inequality
5. Phrases that indicate inequalities
### Practice Makes Perfect
Graph Inequalities on the Number Line
In the following exercises, graph each inequality on the number line and write in interval notation.
Solve Linear Inequalities
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, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
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 graph the solution on the number line and write the solution in interval notation.
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.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not? |
# Solving Linear Equations
## Solve Compound Inequalities
### Solve Compound Inequalities with “and”
Now that we know how to solve linear inequalities, the next step is to look at compound inequalities. A compound inequality is made up of two inequalities connected by the word “and” or the word “or.” For example, the following are compound inequalities.
To solve a compound inequality means to find all values of the variable that make the compound inequality a true statement. We solve compound inequalities using the same techniques we used to solve linear inequalities. We solve each inequality separately and then consider the two solutions.
To solve a compound inequality with the word “and,” we look for all numbers that make both inequalities true. To solve a compound inequality with the word “or,” we look for all numbers that make either inequality true.
Let’s start with the compound inequalities with “and.” Our solution will be the numbers that are solutions to both inequalities known as the intersection of the two inequalities. Consider the intersection of two streets—the part where the streets overlap—belongs to both streets.
To find the solution of the compound inequality, we look at the graphs of each inequality and then find the numbers that belong to both graphs—where the graphs overlap.
For the compound inequality and we graph each inequality. We then look for where the graphs “overlap”. The numbers that are shaded on both graphs, will be shaded on the graph of the solution of the compound inequality. See .
We can see that the numbers between and are shaded on both of the first two graphs. They will then be shaded on the solution graph.
The number is not shaded on the first graph and so since it is not shaded on both graphs, it is not included on the solution graph.
The number two is shaded on both the first and second graphs. Therefore, it is be shaded on the solution graph.
This is how we will show our solution in the next examples.
Sometimes we have a compound inequality that can be written more concisely. For example, and can be written simply as and then we call it a double inequality. The two forms are equivalent.
To solve a double inequality we perform the same operation on all three “parts” of the double inequality with the goal of isolating the variable in the center.
When written as a double inequality, it is easy to see that the solutions are the numbers caught between one and five, including one, but not five. We can then graph the solution immediately as we did above.
Another way to graph the solution of is to graph both the solution of and the solution of We would then find the numbers that make both inequalities true as we did in previous examples.
### Solve Compound Inequalities with “or”
To solve a compound inequality with “or”, we start out just as we did with the compound inequalities with “and”—we solve the two inequalities. Then we find all the numbers that make either inequality true.
Just as the United States is the union of all of the 50 states, the solution will be the union of all the numbers that make either inequality true. To find the solution of the compound inequality, we look at the graphs of each inequality, find the numbers that belong to either graph and put all those numbers together.
To write the solution in interval notation, we will often use the union symbol, to show the union of the solutions shown in the graphs.
### Solve Applications with Compound Inequalities
Situations in the real world also involve compound inequalities. We will use the same problem solving strategy that we used to solve linear equation and inequality applications.
Recall the problem solving strategies are to first read the problem and make sure all the words are understood. Then, identify what we are looking for and assign a variable to represent it. Next, restate the problem in one sentence to make it easy to translate into a compound inequality. Last, we will solve the compound inequality.
### Key Concepts
1. How to solve a compound inequality with “and”
2. Double Inequality
3. How to solve a compound inequality with “or”
### Practice Makes Perfect
Solve Compound Inequalities with “and”
In the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
Solve Compound Inequalities with “or”
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.
Solve Applications with Compound 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? |
# Solving Linear Equations
## Solve Absolute Value Inequalities
### Solve Absolute Value Equations
As we prepare to solve absolute value equations, we review our definition of absolute value.
We learned that both a number and its opposite are the same distance from zero on the number line. Since they have the same distance from zero, they have the same absolute value. For example:
is 5 units away from 0, so
is 5 units away from 0, so
illustrates this idea.
For the equation we are looking for all numbers that make this a true statement. We are looking for the numbers whose distance from zero is 5. We just saw that both 5 and are five units from zero on the number line. They are the solutions to the equation.
The solution can be simplified to a single statement by writing This is read, “x is equal to positive or negative 5”.
We can generalize this to the following property for absolute value equations.
To solve an absolute value equation, we first isolate the absolute value expression using the same procedures we used to solve linear equations. Once we isolate the absolute value expression we rewrite it as the two equivalent equations.
The steps for solving an absolute value equation are summarized here.
Remember, an absolute value is always positive!
Some of our absolute value equations could be of the form where u and v are algebraic expressions. For example,
How would we solve them? If two algebraic expressions are equal in absolute value, then they are either equal to each other or negatives of each other. The property for absolute value equations says that for any algebraic expression, u, and a positive real number, a, if then or
This tells us that
This leads us to the following property for equations with two absolute values.
When we take the opposite of a quantity, we must be careful with the signs and to add parentheses where needed.
### Solve Absolute Value Inequalities with “Less Than”
Let’s look now at what happens when we have an absolute value inequality. Everything we’ve learned about solving inequalities still holds, but we must consider how the absolute value impacts our work.
Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line. For the equation we saw that both 5 and are five units from zero on the number line. They are the solutions to the equation.
What about the inequality Where are the numbers whose distance is less than or equal to 5? We know and 5 are both five units from zero. All the numbers between and 5 are less than five units from zero. See .
In a more general way, we can see that if then See .
This result is summarized here.
After solving an inequality, it is often helpful to check some points to see if the solution makes sense. The graph of the solution divides the number line into three sections. Choose a value in each section and substitute it in the original inequality to see if it makes the inequality true or not. While this is not a complete check, it often helps verify the solution.
### Solve Absolute Value Inequalities with “Greater Than”
What happens for absolute value inequalities that have “greater than”? Again we will look at our definition of absolute value. The absolute value of a number is its distance from zero on the number line.
We started with the inequality We saw that the numbers whose distance is less than or equal to five from zero on the number line were and 5 and all the numbers between and 5. See .
Now we want to look at the inequality Where are the numbers whose distance from zero is greater than or equal to five?
Again both and 5 are five units from zero and so are included in the solution. Numbers whose distance from zero is greater than five units would be less than and greater than 5 on the number line. See .
In a more general way, we can see that if then or See .
This result is summarized here.
### Solve Applications with Absolute Value
Absolute value inequalities are often used in the manufacturing process. An item must be made with near perfect specifications. Usually there is a certain tolerance of the difference from the specifications that is allowed. If the difference from the specifications exceeds the tolerance, the item is rejected.
### Key Concepts
1. Absolute Value
The absolute value of a number is its distance from 0 on the number line.
The absolute value of a number n is written as and for all numbers.
Absolute values are always greater than or equal to zero.
2. Absolute Value Equations
For any algebraic expression, u, and any positive real number, a,
Remember that an absolute value cannot be a negative number.
3. How to Solve Absolute Value Equations
4. Equations with Two Absolute Values
For any algebraic expressions, u and v,
5. Absolute Value Inequalities with or
For any algebraic expression, u, and any positive real number, a,
6. How To Solve Absolute Value Inequalities with or
7. Absolute Value Inequalities with or
For any algebraic expression, u, and any positive real number, a,
8. How To Solve Absolute Value Inequalities with or
### Section Exercises
### Practice Makes Perfect
Solve Absolute Value Equations
In the following exercises, solve.
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
In the following exercises, solve. For each inequality, also graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.
### 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 Review Exercises
### Use a General Strategy to Solve Linear Equations
Solve Equations Using the General Strategy for Solving Linear Equations
In the following exercises, determine whether each number is a solution to the equation.
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 Fraction or Decimal Coefficients
In the following exercises, solve each equation.
### Use a Problem-Solving Strategy
Use a Problem Solving Strategy for Word Problems
In the following exercises, solve using the problem solving strategy for word problems.
Solve Number Word Problems
In the following exercises, solve each number word problem.
Solve Percent Applications
In the following exercises, translate and solve.
In the following exercises, solve.
Solve Simple Interest Applications
In the following exercises, solve.
### Solve a formula for a Specific Variable
Solve a Formula for a Specific Variable
In the following exercises, solve the formula for the specified variable.
Use Formulas to Solve Geometry Applications
In the following exercises, solve using a geometry formula.
### Solve Mixture and Uniform Motion Applications
Solve Coin Word Problems
In the following exercises, solve.
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.
Solve Uniform Motion Applications
In the following exercises, solve.
### Solve Linear Inequalities
Graph Inequalities on the Number Line
In the following exercises, graph the inequality on the number line and write in interval notation.
Solve Linear Inequalities
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Translate Words 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.
Solve Applications with Linear Inequalities
In the following exercises, solve.
### Solve Compound Inequalities
Solve Compound Inequalities with “and”
In each of the following exercises, solve each inequality, graph the solution, and write the solution in interval notation.
Solve Compound Inequalities with “or”
In the following exercises, solve each inequality, graph the solution on the number line, and write the solution in interval notation.
Solve Applications with Compound Inequalities
In the following exercises, solve.
### Solve Absolute Value Inequalities
Solve Absolute Value Equations
In the following exercises, solve.
Solve Absolute Value Inequalities with “less than”
In the following exercises, solve each inequality. Graph the solution and write the solution in interval notation.
Solve Absolute Value Inequalities with “greater than”
In the following exercises, solve. Graph the solution and write the solution in interval notation.
Solve Applications with Absolute Value
In the following exercises, solve.
### Practice Test
In the following exercises, solve each equation.
In the following exercises, graph the inequality 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. |
# Graphs and Functions
## Introduction
Imagine visiting a faraway city or even outer space from the comfort of your living room. It could be possible using virtual reality. This technology creates realistic images that make you feel as if you are truly immersed in the scene and even enable you to interact with them. It is being developed for fun applications, such as video games, but also for architects to plan buildings, car companies to design prototypes, the military to train, and medical students to learn.
Developing virtual reality devices requires modeling the environment using graphs and mathematical relationships. In this chapter, you will graph different relationships and learn ways to describe and analyze graphs. |
# Graphs and Functions
## Graph Linear Equations in Two Variables
### 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 rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. 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 x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that 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 See .
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
When one of the coordinate is zero, the point lies on one of the axes. In the point is on the y-axis and the point is on the x-axis.
The signs of the x-coordinate and y-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:
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. But equations can have more than one variable. Equations with two variables may be of the form An equation of this form is called a linear equation in two variables.
Here is an example of a linear equation in two variables, x and y.
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 of a linear equation.
Most people prefer to have A, B, and C 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 x there is a corresponding y 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 x and y 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.
Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.
A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that 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. Points not on the line are not solutions!
### Graph a Linear Equation by Plotting Points
There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.
The steps to take when graphing a linear equation by plotting points are summarized here.
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 these illustrations.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for x. But the arithmetic is easier if we make “good” choices for the values of x. This way we will avoid fractional answers, which are hard to graph precisely.
### Graph Vertical and Horizontal Lines
Some linear equations have only one variable. They may have just x and no y, or just y without an x. This changes how we make a table of values to get the points to plot.
Let’s consider the equation This equation has only one variable, x. The equation says that x is always equal to so its value does not depend on y. No matter what is the value of y, the value of x is always
So to make a table of values, write in for all the x-values. Then choose any values for y. Since x does not depend on y, 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 the table and connect them with a straight line. Notice that we have graphed a vertical line.
What if the equation has y but no x? Let’s graph the equation This time the y-value is a constant, so in this equation, y does not depend on x. Fill in 4 for all the y’s in and then choose any values for x. We’ll use 0, 2, and 4 for the x-coordinates.
In this figure, we have graphed a horizontal line passing through the y-axis at 4.
What is the difference between the equations and
The equation has both x and y. The value of y depends on the value of x, so the y -coordinate changes according to the value of x. The equation has only one variable. The value of y is constant, it does not depend on the value of x, so the y-coordinate is always 4.
Notice, in the graph, the equation gives a slanted line, while gives a horizontal line.
### Find x- and y-intercepts
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 a line.
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the x-axis. See .
Now, let’s look at the points where these lines cross the y-axis.
Do you see a pattern?
For each line, 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 x-intercept of the line. The x-intercept occurs when y is zero.
In each line, 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 x is zero.
Recognizing that the x-intercept 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 y-intercept, 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 here.
When the line passes through the origin, the x-intercept and the y-intercept are the same point.
### Key Concepts
1. Points on the Axes
2. Quadrant
3. Graph of a Linear Equation: The graph of a linear equation is a straight line.
Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line.
4. How to graph a linear equation by plotting points.
5.
6. Find the
7. How to graph a linear equation using the intercepts.
### 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, 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.
Find
In the following exercises, find the x- and y-intercepts on each graph.
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Mixed Practice
In the following exercises, graph each equation.
### 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 address 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 and Functions
## Slope of a Line
### Find the 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.
In mathematics, the measure of the steepness of a line is called the slope of the line.
The concept of slope has many applications in the real world. In construction the pitch of a roof, the slant of the plumbing pipes, and the steepness of the stairs are all applications of slope. and as you ski or jog down a hill, you definitely experience slope.
We can assign a numerical value to the slope of a line by finding the ratio of the rise and run. The rise is the amount the vertical distance changes while the run measures the horizontal change, as shown in this illustration. Slope is a rate of change. See .
To find the slope of a line, we locate two points on the line whose coordinates are integers. Then we sketch a right triangle where the two points are vertices and one side is horizontal and one side is vertical.
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,
How do we find the slope of horizontal and vertical lines? To find the slope of the horizontal line, we could graph the line, find two points on it, and count the rise and the run. Let’s see what happens when we do this, as shown in the graph below.
Let’s also consider a vertical line, the line as shown in the graph.
The slope is undefined since division by zero is undefined. So we say that the slope of the vertical line is undefined.
All horizontal lines have slope 0. When the y-coordinates are the same, the rise is 0.
The slope of any vertical line is undefined. When the x-coordinates of a line are all the same, the run is 0.
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.
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 as shown in this graph.
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.
The slope is:
### 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.
We can also graph a line 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.
### Graph a Line Using its Slope and Intercept
We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using one point and the slope of the line. 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.
See . Let’s look at the graph of the equation and find its slope and y-intercept.
The red lines in the graph show us the rise is 1 and the run is 2. Substituting into the slope formula:
The y-intercept is
Look at the equation of this line.
Look at the slope and y-intercept.
When a linear equation is solved for y, the coefficient of the x 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.”
Let’s practice finding the values of the slope and y-intercept from the equation of a line.
We have graphed a line using the slope and a point. Now that we know how to find the slope and y-intercept of a line from its equation, we can use the y-intercept as the point, and then count out the slope from there.
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.
### 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. Let’s look for some patterns to help determine the most convenient method to graph a line.
Here are five 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 x and y 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 x or y.
Equation #5 is 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 uses real-world data, different letters are used for the variables, instead of using only x and y. The variable names remind us of what quantities are being measured.
Also, we often will need to extend the axes in our rectangular coordinate system to bigger positive and negative numbers to accommodate the data in the application.
The cost of running some types of 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 and Perpendicular Lines
Two lines that have the same slope are called parallel lines. Parallel lines have the same steepness and 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 .
Verify that both lines have the same slope, and different y-intercepts.
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, like the lines shown in this graph.
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.
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.
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 opposite reciprocals. If the product of the slopes is the lines are perpendicular.
### Key Concepts
1. Slope of a Line
2. How to find the slope of a line from its graph using
3. Slope of a line between two points.
4. How to graph a line given a point and the slope.
5. Slope Intercept Form of an Equation of a Line
6. Parallel Lines
7. Perpendicular Lines
### Practice Makes Perfect
Find the Slope of a Line
In the following exercises, find the slope of each line shown.
In the following exercises, find the slope of each line.
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.
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
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 and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.
### 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? |
# Graphs and Functions
## Find the Equation of a Line
How do online companies 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.
The physical sciences, social sciences, and the business world are full of situations that can be modeled with linear equations relating two variables. 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 is 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 m 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 know the slope and at least one point. Then, we will rewrite the equation in slope-intercept form. Most applications of linear equations use the the slope-intercept form.
We list the steps for easy reference.
### 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.
So far, we have two options for finding an equation of a line: slope-intercept or point-slope. When we start with two points, it makes 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.
The steps are summarized here.
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.
### 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.
This graph shows 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 m. (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, as shown in the graph.
Do the lines appear parallel? Does the second line pass through
We were asked to graph the line, 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.
This graph shows 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 m. (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
We were asked to graph the line, 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 This graph shows us the line and the point
We know every line perpendicular to a vertical 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. How to find an equation of a line given the slope and a point.
2. How to find an equation of a line given two points.
3. How to find an equation of a line parallel to a given line.
4. How 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.
### 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 and Functions
## Graph Linear Inequalities in Two Variables
### Verify Solutions to an Inequality in Two Variables
Previously we learned to solve inequalities with only one variable. We will now learn about inequalities containing two variables. In particular we will look at linear inequalities in two variables which are very similar to linear equations in two variables.
Linear 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 made a profit.
Recall that an inequality with one variable had many solutions. For example, 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, linear inequalities in two variables have many solutions. Any ordered pair that makes an inequality true when we substitute in the values is a solution to a linear 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 shown previously 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 .
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 a is included in the solution:
Similarly, for an inequality in two variables, the boundary line is shown with a solid or dashed line to show whether or not it the line is included in the solution.
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, as shown in the graph. 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 but not a solution to the inequality So the point is on the boundary line.
Let’s take another point above the boundary line and test whether or not it is a solution to the inequality The point clearly looks to above the boundary line, doesn’t it? Is it a solution to the inequality?
So, is a solution to
Any point you choose above the boundary line is a solution to the inequality All points above the boundary line are solutions.
Similarly, all points below the boundary line, the side with and are not solutions to as shown in .
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 in Two Variables
Now that we know what the graph of a linear inequality looks like and how it relates to a boundary equation we can use this knowledge to graph a given linear inequality.
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.
Recall that:
### Solve Applications using Linear Inequalities in Two Variables
Many fields use linear inequalities to model a problem. While our examples may be about simple situations, they give us an opportunity to build our skills and to get a feel for how they might be used.
### Key Concepts
1. How to graph a linear inequality in two variables.
### 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 Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables
### 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 and Functions
## Relations and Functions
### Find the Domain and Range of a Relation
As we go about our daily lives, we have many data items or quantities that are paired to our names. Our social security number, student ID number, email address, phone number and our birthday are matched to our name. There is a relationship between our name and each of those items.
When your professor gets her class roster, the names of all the students in the class are listed in one column and then the student ID number is likely to be in the next column. If we think of the correspondence as a set of ordered pairs, where the first element is a student name and the second element is that student’s ID number, we call this a relation.
The set of all the names of the students in the class is called the domain of the relation and the set of all student ID numbers paired with these students is the range of the relation.
There are many similar situations where one variable is paired or matched with another. The set of ordered pairs that records this matching is a relation.
A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is the relation. The set of all x-coordinates is the domain of the relation and the set of all y-coordinates is the range. Generally we write the numbers in ascending order for both the domain and range.
### Determine if a Relation is a Function
A special type of relation, called a function, occurs extensively in mathematics. A function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
The birthday example from helps us understand this definition. Every person has a birthday but no one has two birthdays. It is okay for two people to share a birthday. It is okay that Danny and Stephen share July 24th as their birthday and that June and Liz share August 2nd. Since each person has exactly one birthday, the relation in is a function.
The relation shown by the graph in includes the ordered pairs and Is that okay in a function? No, as this is like one person having two different birthdays.
In algebra, more often than not, functions will be represented by an equation. It is easiest to see if the equation is a function when it is solved for y. If each value of x results in only one value of y, then the equation defines a function.
### Find the Value of a Function
It is very convenient to name a function and most often we name it f, g, h, F, G, or H. In any function, for each x-value from the domain we get a corresponding y-value in the range. For the function f, we write this range value y as This is called function notation and is read f of x or the value of f at x. In this case the parentheses does not indicate multiplication.
We call x the independent variable as it can be any value in the domain. We call y the dependent variable as its value depends on x.
Much as when you first encountered the variable x, function notation may be rather unsettling. It seems strange because it is new. You will feel more comfortable with the notation as you use it.
Let’s look at the equation To find the value of y when we know to substitute into the equation and then simplify.
The value of the function at is 3.
We do the same thing using function notation, the equation can be written as To find the value when we write:
The value of the function at is 3.
This process of finding the value of for a given value of x is called evaluating the function.
In the last example, we found for a constant value of x. In the next example, we are asked to find with values of x that are variables. We still follow the same procedure and substitute the variables in for the x.
Many everyday situations can be modeled using functions.
### Key Concepts
1. Function Notation: For the function
2. Independent and Dependent Variables: For the function
### Practice Makes Perfect
Find the Domain and Range of a Relation
In the following exercises, for each relation ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercises, use the mapping of the relation to ⓐ list the ordered pairs of the relation, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function, ⓑ find the domain of the relation, and ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function, ⓑ find the domain of the function, and ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function: ⓐ ⓑ ⓒ
In the following exercises, evaluate the function: ⓐ ⓑ ⓒ
In the following exercises, evaluate the function.
In the following exercises, 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? |
# Graphs and Functions
## Graphs of Functions
### Use the Vertical Line Test
In the last section we learned how to determine if a relation is a function. The relations we looked at were expressed as a set of ordered pairs, a mapping or an equation. We will now look at how to tell if a graph is that of a function.
An ordered pair is a solution of a linear equation, if the equation is a true statement when the x- and y-values of the ordered pair are substituted into the equation.
The graph of a linear equation is a straight line where every point on the line is a solution of the equation and every solution of this equation is a point on this line.
In , we can see that, in graph of the equation for every x-value there is only one y-value, as shown in the accompanying table.
A relation is a function if every element of the domain has exactly one value in the range. So the relation defined by the equation is a function.
If we look at the graph, each vertical dashed line only intersects the line at one point. This makes sense as in a function, for every x-value there is only one y-value.
If the vertical line hit the graph twice, the x-value would be mapped to two y-values, and so the graph would not represent a function.
This leads us to the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. If any vertical line intersects the graph in more than one point, the graph does not represent a function.
### Identify Graphs of Basic Functions
We used the equation and its graph as we developed the vertical line test. We said that the relation defined by the equation is a function.
We can write this as in function notation as It still means the same thing. The graph of the function is the graph of all ordered pairs where So we can write the ordered pairs as It looks different but the graph will be the same.
Compare the graph of previously shown in with the graph of shown in . Nothing has changed but the notation.
As we move forward in our study, it is helpful to be familiar with the graphs of several basic functions and be able to identify them.
Through our earlier work, we are familiar with the graphs of linear equations. The process we used to decide if is a function would apply to all linear equations. All non-vertical linear equations are functions. Vertical lines are not functions as the x-value has infinitely many y-values.
We wrote linear equations in several forms, but it will be most helpful for us here to use the slope-intercept form of the linear equation. The slope-intercept form of a linear equation is In function notation, this linear function becomes where m is the slope of the line and b is the y-intercept.
The domain is the set of all real numbers, and the range is also the set of all real numbers.
We will use the graphing techniques we used earlier, to graph the basic functions.
The next function whose graph we will look at is called the constant function and its equation is of the form where b is any real number. If we replace the with y, we get We recognize this as the horizontal line whose y-intercept is b. The graph of the function is also the horizontal line whose y-intercept is b.
Notice that for any real number we put in the function, the function value will be b. This tells us the range has only one value, b.
The identity function, is a special case of the linear function. If we write it in linear function form, we see the slope is 1 and the y-intercept is 0.
The next function we will look at is not a linear function. So the graph will not be a line. The only method we have to graph this function is point plotting. Because this is an unfamiliar function, we make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the square function. We call this graph a parabola. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is not all real numbers. Notice the graph consists of values of y never go below zero. This makes sense as the square of any number cannot be negative. So, the range of the square function is all non-negative real numbers.
The next function we will look at is also not a linear function so the graph will not be a line. Again we will use point plotting, and make sure to choose several positive and negative values as well as 0 for our x-values.
Looking at the result in , we can summarize the features of the cube function. As we consider the domain, notice any real number can be used as an x-value. The domain is all real numbers.
The range is all real numbers. This makes sense as the cube of any non-zero number can be positive or negative. So, the range of the cube function is all real numbers.
The next function we will look at does not square or cube the input values, but rather takes the square root of those values.
Let’s graph the function and then summarize the features of the function. Remember, we can only take the square root of non-negative real numbers, so our domain will be the non-negative real numbers.
Our last basic function is the absolute value function, Keep in mind that the absolute value of a number is its distance from zero. Since we never measure distance as a negative number, we will never get a negative number in the range.
### Read Information from a Graph of a Function
In the sciences and business, data is often collected and then graphed. The graph is analyzed, information is obtained from the graph and then often predictions are made from the data.
We will start by reading the domain and range of a function from its graph.
Remember the domain is the set of all the x-values in the ordered pairs in the function. To find the domain we look at the graph and find all the values of x that have a corresponding value on the graph. Follow the value x up or down vertically. If you hit the graph of the function then x is in the domain.
Remember the range is the set of all the y-values in the ordered pairs in the function. To find the range we look at the graph and find all the values of y that have a corresponding value on the graph. Follow the value y left or right horizontally. If you hit the graph of the function then y is in the range.
We are now going to read information from the graph that you may see in future math classes.
### Key Concepts
1. Vertical Line Test
2. Graph of a Function
3. Linear Function
4. Constant Function
5. Identity Function
6. Square Function
7. Cube Function
8. Square Root Function
9. Absolute Value Function
### Section Exercises
### Practice Makes Perfect
Use the Vertical Line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation.
In the following exercises, use the graph of the function to find the indicated values.
### 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 Review Exercises
### Graph Linear Equations in Two Variables
Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system.
In the following exercises, determine which ordered pairs are solutions to the given equations.
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.
Find
In the following exercises, find the x- and y-intercepts.
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
Find the Slope of a Line
In the following exercises, find the slope of each line shown.
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.
Graph a Line Using Its Slope and Intercept
In the following exercises, identify the slope and y-intercept of each line.
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 and Perpendicular Lines
In the following exercises, use slopes and y-intercepts to determine if the lines are parallel, perpendicular, or neither.
### 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 in Two Variables
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 Two Variables
In the following exercises, graph each linear inequality.
Solve Applications using Linear Inequalities in Two Variables
### Relations and Functions
Find the Domain and Range of a Relation
In the following exercises, for each relation, ⓐ find the domain of the relation ⓑ find the range of the relation.
In the following exercise, use the mapping of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercise, use the graph of the relation to ⓐ list the ordered pairs of the relation ⓑ find the domain of the relation ⓒ find the range of the relation.
Determine if a Relation is a Function
In the following exercises, use the set of ordered pairs to ⓐ determine whether the relation is a function ⓑ find the domain of the relation ⓒ find the range of the relation.
In the following exercises, use the mapping to ⓐ determine whether the relation is a function ⓑ find the domain of the function ⓒ find the range of the function.
In the following exercises, determine whether each equation is a function.
Find the Value of a Function
In the following exercises, evaluate the function:
ⓐ ⓑ ⓒ
In the following exercises, evaluate the function.
### Graphs of Functions
Use the Vertical line Test
In the following exercises, determine whether each graph is the graph of a function.
Identify Graphs of Basic Functions
In the following exercises, ⓐ graph each function ⓑ state its domain and range. Write the domain and range in interval notation.
Read Information from a Graph of a Function
In the following exercises, use the graph of the function to find its domain and range. Write the domain and range in interval notation
In the following exercises, use the graph of the function to find the indicated values.
### 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.
In the following exercises, ⓐ graph each function ⓑ state its domain and range.Write the domain and range in interval notation. |
# Systems of Linear Equations
## Introduction
Climb into your car. Put on your seatbelt. Choose your destination and then…relax. That’s right. You don’t have to do anything else because you are in an autonomous car, or one that navigates its way to your destination! No cars are fully autonomous at the moment and so you theoretically still need to have your hands on the wheel. Self-driving cars may help ease traffic congestion, prevent accidents, and lower pollution. The technology is thanks to computer programmers who are developing software to control the navigation of the car. These programmers rely on their understanding of mathematics, including relationships between equations. In this chapter, you will learn how to solve systems of linear equations in different ways and use them to analyze real-world situations. |
# Systems of Linear Equations
## Solve Systems of Linear Equations with Two Variables
### Determine Whether an Ordered Pair is a Solution of a System of Equations
In Solving Linear Equations, we learned how to solve linear equations with one variable. Now we will work with two or more linear equations grouped together, which is known as a system of linear equations.
In this section, we will focus our work on systems of two linear equations in two unknowns. We will solve larger systems of equations later in this chapter.
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, such as 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 that make both equations true. These are called the solutions of 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.
### Solve a System of Linear Equations by Graphing
In this section, 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.
Each time we demonstrate a new method, we will use it on the same system of linear equations. At the end of the section 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 here.
In the next example, we’ll first re-write the equations into slope–intercept form as this will make it easy for us to quickly graph the lines.
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.
Sometimes the equations in a system represent the same line. Since every point on the line makes both equations true, there are infinitely many ordered pairs that make both equations true. There are infinitely many solutions to the system.
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.
The systems of equations in and each had two intersecting lines. Each system had one solution.
In , the equations gave coincident lines, and so the system had infinitely many solutions.
The systems in those three examples had at least one solution. A system of equations that has at least one solution is called a consistent system.
A system with parallel lines, like , has no solution. We call a system of equations like this inconsistent. It has no solution.
We also categorize the equations in a system of equations by calling the equations independent or dependent. If two equations are independent, 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 below and .
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 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.
### Solve a System of Equations by Substitution
We will now solve systems of linear equations by the substitution method.
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.
Be very careful with the signs in the next example.
### Solve a System 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.
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 we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by
Then rewrite the system of equations.
Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations.
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 here for easy reference.
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 the LCD of all the fractions in the equation.
When we solved the system 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.
The same is true using substitution or elimination. If the equation at the end of substitution or elimination is a true statement, we have a consistent but dependent system and the system of equations has infinitely many solutions. If the equation at the end of substitution or elimination is a false statement, we have an inconsistent system and the system of equations has no solution.
### Choose the Most Convenient Method to Solve a System of Linear Equations
When you solve a system of linear equations in in an application, you will 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. How to solve a system of linear equations by graphing.
2. How to solve a system of equations by substitution.
3. How to solve a system of equations by elimination.
### 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.
Without graphing, determine the number of solutions and then classify the system of equations.
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination.
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.
### 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
## Solve Applications with Systems of Equations
### Solve Direct Translation Applications
Systems of linear equations are very useful for solving applications. Some people find setting up word problems with two variables easier than setting them up with just one variable. To solve an application, we’ll first translate the words into a system of linear equations. Then we will decide the most convenient method to use, and then solve the system.
We solved number problems with one variable earlier. Let’s see how differently it works using two variables.
As you solve each application, remember to analyze which method of solving the system of equations would be most convenient.
### Solve Geometry Applications
We will now solve geometry applications using systems of linear equations. We will need to add complementary angles and supplementary angles 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.
In the next example, we remember that the measures of supplementary angles add to 180.
Recall that the angles of a triangle add up to 180 degrees. A right triangle has one angle that is 90 degrees. What does that tell us about the other two angles? In the next example we will be finding the measures of the other two angles.
Often it is helpful when solving geometry applications to draw a picture to visualize the situation.
### 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 where D is the distance traveled, 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.
The images below 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.
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
Now, 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 the next example.
Wind currents affect airplane speeds in the same way as water currents affect boat speeds. We’ll see this in the next example. 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.
### Key Concepts
1. How To Solve Applications with Systems of Equations
### Practice Makes Perfect
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.
### 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 Mixture Applications with Systems of Equations
### Solve Mixture Applications
Mixture application involve combining two or more quantities. 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 the next example, 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 simple interest applications is 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 the next example.
The next example requires that we find the principal given the amount of interest earned.
### Solve applications of cost and revenue functions
Suppose a company makes and sells x units of a product. The cost to the company is the total costs to produce x units. This is the cost to manufacture for each unit times x, the number of units manufactured, plus the fixed costs.
The revenue is the money the company brings in as a result of selling x units. This is the selling price of each unit times the number of units sold.
When the costs equal the revenue we say the business has reached the break-even point.
### Key Concepts
1. Cost function: The cost function is the cost to manufacture each unit times x, the number of units manufactured, plus the fixed costs.
2. Revenue: The revenue function is the selling price of each unit times x, the number of units sold.
3. Break-even point: The break-even point is when the revenue equals the costs.
### 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.
Solve Applications of Cost and Revenue Functions
### 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 Systems of Equations with Three Variables
### Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables
In this section, we will extend our work of solving a system of linear equations. So far we have worked with systems of equations with two equations and two variables. Now we will work with systems of three equations with three variables. But first let's review what we already know about solving equations and systems involving up to two variables.
We learned earlier that the graph of a linear equation, is a line. Each point on the line, an ordered pair is a solution to the equation. For a system of two equations with two variables, we graph two lines. Then we can see that all the points that are solutions to each equation form a line. 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
We know 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.
Similarly, for a linear equation with three variables every solution to the equation is an ordered triple, , that makes the equation true.
All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we’ll find the solution to the system.
When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.
To solve a system of three linear equations, we want to find the values of the variables that are solutions to all three equations. In other words, we are looking for the ordered triple that makes all three equations true. These are called the solutions of the system of three linear equations with three variables.
To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.
### Solve a System of Linear Equations with Three Variables
To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!
Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.
The steps are summarized here.
When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.
When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.
### Solve Applications using Systems of Linear Equations with Three Variables
Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.
### Key Concepts
1. Linear Equation in Three Variables: A linear equation with three variables, where a, b, c, and d are real numbers and a, b, and c are not all 0, is of the form
Every solution to the equation is an ordered triple, that makes the equation true.
2. How to solve a system of linear equations with three variables.
### Practice Makes Perfect
Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables
In the following exercises, determine whether the ordered triple is a solution to the system.
Solve a System of Linear Equations with Three Variables
In the following exercises, solve the system of equations.
Solve Applications using Systems of Linear Equations with Three Variables
In the following exercises, solve the given problem.
### 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 Systems of Equations Using Matrices
### Write the Augmented Matrix for a System of Equations
Solving a system of equations can be a tedious operation where a simple mistake can wreak havoc on finding the solution. An alternative method which uses the basic procedures of elimination but with notation that is simpler is available. The method involves using a matrix. A matrix is a rectangular array of numbers arranged in rows and columns.
We will use a matrix to represent a system of linear equations. We write each equation in standard form and the coefficients of the variables and the constant of each equation becomes a row in the matrix. Each column then would be the coefficients of one of the variables in the system or the constants. A vertical line replaces the equal signs. We call the resulting matrix the augmented matrix for the system of equations.
Notice the first column is made up of all the coefficients of x, the second column is the all the coefficients of y, and the third column is all the constants.
It is important as we solve systems of equations using matrices to be able to go back and forth between the system and the matrix. The next example asks us to take the information in the matrix and write the system of equations.
### Use Row Operations on a Matrix
Once a system of equations is in its augmented matrix form, we will perform operations on the rows that will lead us to the solution.
To solve by elimination, it doesn’t matter which order we place the equations in the system. Similarly, in the matrix we can interchange the rows.
When we solve by elimination, we often multiply one of the equations by a constant. Since each row represents an equation, and we can multiply each side of an equation by a constant, similarly we can multiply each entry in a row by any real number except 0.
In elimination, we often add a multiple of one row to another row. In the matrix we can replace a row with its sum with a multiple of another row.
These actions are called row operations and will help us use the matrix to solve a system of equations.
Performing these operations is easy to do but all the arithmetic can result in a mistake. If we use a system to record the row operation in each step, it is much easier to go back and check our work.
We use capital letters with subscripts to represent each row. We then show the operation to the left of the new matrix. To show interchanging a row:
To multiply row 2 by :
To multiply row 2 by and add it to row 1:
Now that we have practiced the row operations, we will look at an augmented matrix and figure out what operation we will use to reach a goal. This is exactly what we did when we did elimination. We decided what number to multiply a row by in order that a variable would be eliminated when we added the rows together.
Given this system, what would you do to eliminate x?
This next example essentially does the same thing, but to the matrix.
### Solve Systems of Equations Using Matrices
To solve a system of equations using matrices, we transform the augmented matrix into a matrix in row-echelon form using row operations. For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.
Once we get the augmented matrix into row-echelon form, we can write the equivalent system of equations and read the value of at least one variable. We then substitute this value in another equation to continue to solve for the other variables. This process is illustrated in the next example.
The steps are summarized here.
Here is a visual to show the order for getting the 1’s and 0’s in the proper position for row-echelon form.
We use the same procedure when the system of equations has three equations.
So far our work with matrices has only been with systems that are consistent and independent, which means they have exactly one solution. Let’s now look at what happens when we use a matrix for a dependent or inconsistent system.
The last system was inconsistent and so had no solutions. The next example is dependent and has infinitely many solutions.
### Key Concepts
1. Matrix: A matrix is a rectangular array of numbers arranged in rows and columns. A matrix with m rows and n columns has order The matrix on the left below has 2 rows and 3 columns and so it has order We say it is a 2 by 3 matrix.
Each number in the matrix is called an element or entry in the matrix.
2. Row Operations: In a matrix, the following operations can be performed on any row and the resulting matrix will be equivalent to the original matrix.
3. Row-Echelon Form: For a consistent and independent system of equations, its augmented matrix is in row-echelon form when to the left of the vertical line, each entry on the diagonal is a 1 and all entries below the diagonal are zeros.
4. How to solve a system of equations using matrices.
### Practice Makes Perfect
Write the Augmented Matrix for a System of Equations
In the following exercises, write each system of linear equations as an augmented matrix.
Write the system of equations that corresponds to the augmented matrix.
Use Row Operations on a Matrix
In the following exercises, perform the indicated operations on the augmented matrices.
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
In the following exercises, solve each system of equations using a matrix.
### 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
## Solve Systems of Equations Using Determinants
In this section we will learn of another method to solve systems of linear equations called Cramer’s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.
### Evaluate the Determinant of a Matrix
If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix we first write it as To get the real number value of the determinant we subtract the products of the diagonals, as shown.
### Evaluate the Determinant of a Matrix
To evaluate the determinant of a matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the determinant found by eliminating the row and column in the determinant that contains the entry.
To find the minor of entry we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the determinant that remains.
To find the minor of entry we eliminate the row and column that contain it. So we eliminate the 2nd row and 2nd column. Then we write the determinant that remains.
We are now ready to evaluate a determinant. To do this we expand by minors, which allows us to evaluate the determinant using determinants—which we already know how to evaluate!
To evaluate a determinant by expanding by minors along the first row, we use the following pattern:
Remember, to find the minor of an entry we eliminate the row and column that contains the entry.
To evaluate a determinant we can expand by minors using any row or column. Choosing a row or column other than the first row sometimes makes the work easier.
When we expand by any row or column, we must be careful about the sign of the terms in the expansion. To determine the sign of the terms, we use the following sign pattern chart.
Notice that the sign pattern in the first row matches the signs between the terms in the expansion by the first row.
Since we can expand by any row or column, how do we decide which row or column to use? Usually we try to pick a row or column that will make our calculation easier. If the determinant contains a 0, using the row or column that contains the 0 will make the calculations easier.
### Use Cramer’s Rule to Solve Systems of Equations
Cramer’s Rule is a method of solving systems of equations using determinants. It can be derived by solving the general form of the systems of equations by elimination. Here we will demonstrate the rule for both systems of two equations with two variables and for systems of three equations with three variables.
Let’s start with the systems of two equations with two variables.
Notice that to form the determinant D, we use take the coefficients of the variables.
Notice that to form the determinant and we substitute the constants for the coefficients of the variable we are finding.
To solve a system of three equations with three variables with Cramer’s Rule, we basically do what we did for a system of two equations. However, we now have to solve for three variables to get the solution. The determinants are also going to be which will make our work more interesting!
Cramer’s rule does not work when the value of the D determinant is 0, as this would mean we would be dividing by 0. But when the system is either inconsistent or dependent.
When the value of and and are all zero, the system is consistent and dependent and there are infinitely many solutions.
When the value of and and are not all zero, the system is inconsistent and there is no solution.
In the next example, we will use the values of the determinants to find the solution of the system.
### Solve Applications using Determinants
An interesting application of determinants allows us to test if points are collinear. Three points and are collinear if and only if the determinant below is zero.
We will use this property in the next example.
### Key Concepts
1. Determinant: The determinant of any square matrix where a, b, c, and d are real numbers, is
2. Expanding by Minors along the First Row to Evaluate a 3 × 3 Determinant: To evaluate a determinant by expanding by minors along the first row, the following pattern:
3. Sign Pattern: When expanding by minors using a row or column, the sign of the terms in the expansion follow the following pattern.
4. Cramer’s Rule: For the system of equations the solution can be determined by
Notice that to form the determinant D, we use take the coefficients of the variables.
5. How to solve a system of two equations using Cramer’s rule.
### Practice Makes Perfect
Evaluate the Determinant of a 2 × 2 Matrix
In the following exercises, evaluate the determinant of each square matrix.
Evaluate the Determinant of a 3 × 3 Matrix
In the following exercises, find and then evaluate the indicated minors.
In the following exercises, evaluate each determinant by expanding by minors along the first row.
In the following exercises, evaluate each determinant by expanding by minors.
Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s Rule.
Solve Applications Using Determinants
In the following exercises, determine whether the given points are collinear.
### 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
## 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 here.
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 the next example.
Some systems of linear inequalities where the boundary lines are parallel will have a solution. We’ll see this in the next example.
### 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 we add inequalities to the system as additional requirements.
When we use variables other than x and y to define an unknown quantity, we must change the names of the axes of the graph as well.
### Key Concepts
1. Solutions of a System of Linear Inequalities: Solutions of a system of linear inequalities are the values of the variables that make all the inequalities true. The solution of a system of linear inequalities is shown as a shaded region in the x, y coordinate system that includes all the points whose ordered pairs make the inequalities true.
2. How 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.
### 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 Review Exercises
### Solve Systems of Linear Equations with Two Variables
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.
In the following exercises, without graphing determine the number of solutions and then classify the system of equations.
Solve a System of Equations by Substitution
In the following exercises, solve the systems of equations by substitution.
Solve a System of Equations by Elimination
In the following exercises, solve the systems of equations by elimination
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
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 with Systems of Equations
For the following exercises, translate to a system of equations and solve.
Solve Interest Applications
For the following exercises, translate to a system of equations and solve.
### Solve Systems of Equations with Three Variables
Solve Systems of Equations with Three Variables
In the following exercises, determine whether the ordered triple is a solution to the system.
Solve a System of Linear Equations with Three Variables
In the following exercises, solve the system of equations.
Solve Applications using Systems of Linear Equations with Three Variables
### Solve Systems of Equations Using Matrices
Write the Augmented Matrix for a System of Equations.
Write each system of linear equations as an augmented matrix.
Write the system of equations that that corresponds to the augmented matrix.
In the following exercises, perform the indicated operations on the augmented matrices.
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
### Solve Systems of Equations Using Determinants
Evaluate the Determinant of a 2 × 2 Matrix
In the following exercise, evaluate the determinate of the square matrix.
Evaluate the Determinant of a 3 × 3 Matrix
In the following exercise, find and then evaluate the indicated minors.
In the following exercise, evaluate each determinant by expanding by minors along the first row.
In the following exercise, evaluate each determinant by expanding by minors.
Use Cramer’s Rule to Solve Systems of Equations
In the following exercises, solve each system of equations using Cramer’s rule
Solve Applications Using Determinants
In the following exercises, determine whether the given points are collinear.
### 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.
### Chapter 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.
Solve the system of equations using a matrix.
Solve using Cramer’s rule.
In the following exercises, translate to a system of equations and solve. |
# Polynomials and Polynomial Functions
## Introduction
You may have coins and paper money in your wallet, but you may soon want to acquire a type of currency called bitcoins. They exist only in a digital wallet on your computer. You can use bitcoins to pay for goods at some companies, or save them as an investment. Although the future of bitcoins is uncertain, investment brokers are beginning to investigate ways to make business predictions using this digital currency. Understanding how bitcoins are created and obtained requires an understanding of a type of function known as a polynomial function. In this chapter you will investigate polynomials and polynomial functions and learn how to perform mathematical operations on them. |
# Polynomials and Polynomial Functions
## Add and Subtract Polynomials
### Determine the Degree of Polynomials
We have learned that a term is a constant or the product of a constant and one or more variables. A monomial is an algebraic expression with one term. When it is of the form where a is a constant and m is a whole number, it is called a monomial in one variable. Some examples of monomials in one variable are
and
. Monomials can also have more than one variable such as
and
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.
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.
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.
Let's start by looking at a monomial. The monomial has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, is 3 so the degree is 3.
Here are some additional examples.
Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.
### Add and Subtract Polynomials
We 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 coefficients.
Remember that like terms must have the same variables with the same exponents.
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.
Be careful with the signs as you distribute while subtracting the polynomials in the next example.
To subtract from we write it as placing the first.
When we add and subtract more than two polynomials, the process is the same.
### Evaluate a Polynomial Function for a Given Value
A polynomial function is a function defined by a polynomial. For example, and are polynomial functions, because and are polynomials.
In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.
The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.
### Add and Subtract Polynomial Functions
Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.
### Key Concepts
1. Monomial
2. Polynomials
3. Degree of a Polynomial
### Practice Makes Perfect
Determine the Type of Polynomials
In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.
Add and Subtract Polynomials
In the following exercises, add or subtract the monomials.
In the following exercises, add the polynomials.
In the following exercises, subtract the polynomials.
In the following exercises, subtract the polynomials.
In the following exercises, find the difference of the polynomials.
In the following exercises, add the polynomials.
In the following exercises, add or subtract the polynomials.
Evaluate a Polynomial Function for a Given Value
In the following exercises, find the function values for each polynomial function.
In the following exercises, find the height for each polynomial function.
Add and Subtract Polynomial Functions
In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).
### 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 and Polynomial Functions
## Properties of Exponents and Scientific Notation
### Simplify Expressions Using the Properties for Exponents
Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression the exponent m tells us how many times we use the base a as a factor.
Let’s review the vocabulary for expressions with exponents.
When we combine like terms by adding and subtracting, we 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.
First, we will look at an example that leads to the Product Property.
Notice that 5 is the sum of the exponents, 2 and 3. We see is or
The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.
Now we will look at an exponent property for division. 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. We see is or . We see is or 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. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. . This leads to the Quotient Property for Exponents.
A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like We know 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 We will simplify in two ways to lead us to the definition of the Zero Exponent Property. In general, for
We see simplifies to and to 1. So Any non-zero base raised to the power of zero equals 1.
In this text, we assume any variable that we raise to the zero power is not zero.
### Use the Definition of a Negative Exponent
We saw that the Quotient Property for Exponents 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 see is or
We can also simplify by dividing out common factors:
This implies that and it leads us to the definition of a negative exponent. If n is an integer and then
Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.
This implies and is another form of the definition of Properties of Negative Exponents.
The negative exponent tells us we can rewrite 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.
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.
Now that we have negative exponents, we will use the Product Property with expressions that have negative 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.
Notice the 6 is the product of the exponents, 2 and 3. We see that is or
We multiplied the exponents. This leads to the Power Property for Exponents.
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.
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.
We see that is
This leads to the Quotient to a Power Property for Exponents.
We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.
### Use Scientific Notation
Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.
Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means and 0.004 means
If we write the 1,000 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 ten, 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.
The power of 10 is positive when the number is larger than 1:
The power of 10 is negative when the number is between 0 and 1:
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.
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. Exponential Notation
This is read a to the power.
In the expression , the exponent m tells us how many times we use the base a as a factor.
2. Product Property for Exponents
If a is a real number and m and n are integers, then
To multiply with like bases, add the exponents.
3. Quotient Property for Exponents
If is a real number, and m and n are integers, then
4. Zero Exponent
5. Negative Exponent
6. Quotient to a Negative Exponent Property
If are real numbers, and is an integer, then
7. Power Property for Exponents
If is a real number and are integers, then
To raise a power to a power, multiply the exponents.
8. Product to a Power Property for Exponents
If a and b are real numbers and m is a whole number, then
To raise a product to a power, raise each factor to that power.
9. Quotient to a Power Property for Exponents
If and are real numbers, and is an integer, then
To raise a fraction to a power, raise the numerator and denominator to that power.
10. Summary of Exponent Properties
If a and b are real numbers, and m and n are integers, then
11. Scientific Notation
A number is expressed in scientific notation when it is of the form
12. How to convert a decimal to scientific notation.
13. How to convert scientific notation to decimal form.
### Practice Makes Perfect
Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
In the following exercises, simplify each expression using the Product Property.
In the following exercises, simplify each expression using the Power Property.
In the following exercises, simplify each expression using the Product to a Power Property.
In the following exercises, simplify each expression using the Quotient to a Power Property.
In the following exercises, simplify each expression by applying several properties.
### Mixed Practice
In the following exercises, simplify each expression.
Use Scientific Notation
In the following exercises, write each number in scientific notation.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.
### 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 and Polynomial Functions
## Multiply Polynomials
### Multiply Monomials
We are ready to perform operations on polynomials. Since monomials are algebraic expressions, we can use the properties of exponents to multiply monomials.
### Multiply a Polynomial by a Monomial
Multiplying a polynomial by a monomial is really just applying the Distributive Property.
### 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.
If you multiply binomials often enough you may notice a pattern. Notice that the first term in the result is the product of the first terms in each binomial. The second and third terms are the product of multiplying the two outer terms and then the two inner terms. And the last term results from multiplying the two last terms,
We abbreviate “First, Outer, Inner, Last” as FOIL. The letters stand for ‘First, Outer, Inner, Last’. We use this as another method of multiplying binomials. The word FOIL is easy to remember and ensures we find all four products.
Let’s multiply using both methods.
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.
Now we will do an example where we use the FOIL pattern to multiply two binomials.
The final products in the last example were trinomials because we could combine the two middle terms. This is not always the case.
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 Polynomial by a Polynomial
We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we’re ready to multiply a polynomial by a polynomial. Remember, FOIL will not work in this case, but we can use either the Distributive Property or the Vertical Method.
We have now seen two methods you can use to multiply a polynomial by a polynomial. 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.
### Multiply Special Products
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 multiplying them, there is less work to do if you learn to use a pattern. Let’s start by looking at three examples and look for 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 first term 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 first term of the product, square the first term.
Where did the last term 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 last term of the product, square the last term.
Finally, look at the middle term. 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 middle term of the product, multiply the terms and double their product.
Putting it all together:
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.
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 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.
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.
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.
To get the first term, square the first term.
The last term came from multiplying the last terms, the square of the last term.
To get the last term, square the last term.
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:
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.
### Multiply Polynomial Functions
Just as polynomials can be multiplied, polynomial functions can also be multiplied.
### Key Concepts
1. How to use the FOIL method to multiply two binomials.
2. Multiplying Two Binomials: To multiply binomials, use the:
3. Multiplying a Polynomial by a Polynomial: To multiply a trinomial by a binomial, use the:
4. Binomial Squares Pattern
If a and b are real numbers,
5. Product of Conjugates Pattern
If are real numbers
The product is called a difference of squares.
To multiply conjugates, square the first term, square the last term, write it as a difference of squares.
6. Comparing the Special Product Patterns
7. Multiplication of Polynomial Functions:
### Practice Makes Perfect
Multiply Monomials
In the following exercises, multiply the monomials.
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 Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property; ⓑ the Vertical Method.
Multiply Special Products
In the following exercises, multiply. Use either method.
In the following exercises, square each binomial using the Binomial Squares Pattern.
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
In the following exercises, find each product.
### Mixed Practice
Multiply Polynomial Functions
### 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 and Polynomial Functions
## Dividing Polynomials
### Dividing Monomials
We are now familiar with all the properties of exponents and used them to multiply polynomials. Next, we’ll use these properties to divide monomials and 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.
### Divide a Polynomial by a Monomial
Now that we know how to divide a monomial by a monomial, 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. The sum simplifies to
Now we will do this in reverse to split a single fraction into separate fractions. For example, can be written
This is the “reverse” of fraction addition and it states that if a, b, and c are numbers where then We will use this to divide polynomials by monomials.
### Divide Polynomials Using Long Division
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 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 the next example, 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 previous examples. The terms were written in descending order of degrees, and there were no missing degrees. The dividend in this example will be It is missing an term. We will add in as a placeholder.
In the next example, we will divide by As we divide, we will have to consider the constants as well as the variables.
### Divide Polynomials using Synthetic Division
As we have mentioned before, mathematicians like to find patterns to make their work easier. Since long division can be tedious, let’s look back at the long division we did in and look for some patterns. We will use this as a basis for what is called synthetic division. The same problem in the synthetic division format is shown next.
Synthetic division basically just removes unnecessary repeated variables and numbers. Here all the and are removed. as well as the and as they are opposite the term above.
The first row of the synthetic division is the coefficients of the dividend. The is the opposite of the 5 in the divisor.
The second row of the synthetic division are the numbers shown in red in the division problem.
The third row of the synthetic division are the numbers shown in blue in the division problem.
Notice the quotient and remainder are shown in the third row.
The following example will explain the process.
In the next example, we will do all the steps together.
### Divide Polynomial Functions
Just as polynomials can be divided, polynomial functions can also be divided.
### Use the Remainder and Factor Theorem
Let’s look at the division problems we have just worked that ended up with a remainder. They are summarized in the chart below. If we take the dividend from each division problem and use it to define a function, we get the functions shown in the chart. When the divisor is written as the value of the function at is the same as the remainder from the division problem.
To see this more generally, we realize we can check a division problem by multiplying the quotient times the divisor and add the remainder. In function notation we could say, to get the dividend we multiply the quotient, times the divisor, and add the remainder, r.
This leads us to the Remainder Theorem.
When we divided by in the result was To check our work, we multiply by to get .
Written this way, we can see that and are factors of When we did the division, the remainder was zero.
Whenever a divisor, divides a polynomial function, and resulting in a remainder of zero, we say is a factor of
The reverse is also true. If is a factor of then will divide the polynomial function resulting in a remainder of zero.
We will state this in the Factor Theorem.
### Key Concepts
1. Division of a Polynomial by a Monomial
2. Division of Polynomial Functions
3. Remainder Theorem
4. Factor Theorem: For any polynomial function
### Section Exercises
### Practice Makes Perfect
Divide Monomials
In the following exercises, divide the monomials.
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial.
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if is a factor of the polynomial function.
### 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?
### Chapter Review Exercises
### Add and Subtract Polynomials
Types of Polynomials
In the following exercises, determine the type of polynomial.
Add and Subtract Polynomials
In the following exercises, add or subtract the polynomials.
In the following exercises, simplify.
Evaluate a Polynomial Function for a Given Value of the Variable
In the following exercises, find the function values for each polynomial function.
Add and Subtract Polynomial Functions
In the following exercises, find ⓐ (f + g)(x) ⓑ (f + g)(3) ⓒ (f − g)(x) ⓓ (f − g)(−2)
### Properties of Exponents and Scientific Notation
Simplify Expressions Using the Properties for Exponents
In the following exercises, simplify each expression using the properties for exponents.
Use the Definition of a Negative Exponent
In the following exercises, simplify each expression.
In the following exercises, simplify each expression using the Product Property.
In the following exercises, simplify each expression using the Power Property.
In the following exercises, simplify each expression using the Product to a Power Property.
In the following exercises, simplify each expression using the Quotient to a Power Property.
In the following exercises, simplify each expression by applying several properties.
In the following exercises, write each number in scientific notation.
In the following exercises, convert each number to decimal form.
In the following exercises, multiply or divide as indicated. Write your answer in decimal form.
### Multiply Polynomials
Multiply Monomials
In the following exercises, multiply the monomials.
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 Polynomial by a Polynomial
In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.
In the following exercises, multiply. Use either method.
Multiply Special Products
In the following exercises, square each binomial using the Binomial Squares Pattern.
In the following exercises, multiply each pair of conjugates using the Product of Conjugates.
### Divide Monomials
Divide Monomials
In the following exercises, divide the monomials.
Divide a Polynomial by a Monomial
In the following exercises, divide each polynomial by the monomial
Divide Polynomials using Long Division
In the following exercises, divide each polynomial by the binomial.
Divide Polynomials using Synthetic Division
In the following exercises, use synthetic Division to find the quotient and remainder.
Divide Polynomial Functions
In the following exercises, divide.
Use the Remainder and Factor Theorem
In the following exercises, use the Remainder Theorem to find the remainder.
In the following exercises, use the Factor Theorem to determine if is a factor of the polynomial function.
### Chapter Practice Test
In the following exercises, simplify and write your answer in exponential notation. |
# Factoring
## Introduction to Factoring
An epidemic of a disease has broken out. Where did it start? How is it spreading? What can be done to control it? Answers to these and other questions can be found by scientists known as epidemiologists. They collect data and analyze it to study disease and consider possible control measures. Because diseases can spread at alarming rates, these scientists must use their knowledge of mathematics involving factoring. In this chapter, you will learn how to factor and apply factoring to real-life situations. |
# 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 reverse 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.
We summarize the steps we use to find the greatest common factor.
The next example will show us the steps to find the greatest common factor of three expressions.
### Factor the Greatest Common Factor from a Polynomial
It is sometimes useful to represent a number as a product of factors, for example, 12 as or In algebra, it can also be useful to represent a polynomial in factored form. We will start with a product, such as 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!
When the leading coefficient is negative, we factor the negative out as part of the GCF.
So far our greatest common factors have been monomials. In the next example, the greatest common factor is a binomial.
### Factor by Grouping
Sometimes there is no common factor of all the terms of a polynomial. When there are four terms we separate 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. How to find the greatest common factor (GCF) of two expressions.
2. Distributive Property: If a, b, and c are real numbers, then
The form on the left is used to multiply. The form on the right is used to factor.
3. How to factor the greatest common factor from a polynomial.
4. Factor as a Noun and a Verb: We use “factor” as both a noun and a verb.
5. How to factor by grouping.
### 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.
### 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
### Factor Trinomials of the Form
You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication. To factor the trinomial means to start with the product, and end with the factors.
To figure out how we would factor a trinomial of the form such as and factor it to let’s start with two general binomials of the form and
This tells us that to factor a trinomial of the form we need two factors and where the two numbers m and n multiply to c and add to b.
Let’s summarize the steps we used to find the factors.
In the first example, 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.
How do you get a positive product and a negative sum? We use two negative numbers.
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.
How do you get a negative product and a positive sum? We use one positive and one negative number.
When we factor trinomials, we must have the terms written in descending order—in order from highest degree to lowest degree.
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 How To Factor trinomials.
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 ax2 + bx + c using Trial and Error
Our next step is 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 we’ve used so far. Let’s do an example to see how this works.
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, 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 the factors in the first case will work. Let’s use FOIL to check.
Our result of the factoring is:
Remember, 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 and remember if the leading coefficient is negative, so is the GCF.
### Factor Trinomials of the Form 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!
The “ac” method is summarized here.
Don’t forget to look for a common factor!
### Factor Using Substitution
Sometimes a trinomial does not appear to be in the form. However, we can often make a thoughtful substitution that will allow us to make it fit the form. This is called factoring by substitution. It is standard to use u for the substitution.
In the the middle term has a variable, x, and its square, is the variable part of the first term. Look for this relationship as you try to find a substitution.
Sometimes the expression to be substituted is not a monomial.
### Key Concepts
1. How to factor trinomials of the form
2. Strategy for Factoring Trinomials of the Form When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.
Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.
3. How to factor trinomials of the form
4. How to factor trinomials of the form
### Practice Makes Perfect
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form
In the following exercises, factor each trinomial of the form
Factor Trinomials of the Form
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form
In the following exercises, factor using the ‘ac’ method.
Factor Using Substitution
In the following exercises, factor using substitution.
Mixed Practice
In the following exercises, factor each expression using any method.
### 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? |
# Factoring
## Factor Special Products
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. We squared a binomial using the Binomial Squares pattern in a previous chapter.
The trinomial is called a perfect square trinomial. It is the square of the binomial
In this chapter, 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 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 the product, 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 first step in factoring is to look for a greatest common factor. 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:
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!
The next example shows variables in both terms.
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.
Also, to completely factor the binomial in the next example, we’ll factor a difference of squares twice!
The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.
### 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 will 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 .
In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.
The first term in the next example is a binomial cubed.
### Key Concepts
1. Perfect Square Trinomials Pattern: If a and b are real numbers,
2. How to factor perfect square trinomials.
3. Difference of Squares Pattern: If are real numbers,
4. How to factor differences of squares.
5. Sum and Difference of Cubes Pattern
6. How to factor the sum or difference of cubes.
### Practice Makes Perfect
Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
Mixed Practice
In the following exercises, factor completely.
### 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
## 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. The following chart summarizes all the factoring methods we have covered, and outlines a strategy you should use when factoring polynomials.
Remember, a polynomial is completely factored if, other than monomials, its factors are prime!
Be careful when you are asked to factor a binomial as there are several options!
The next example can be factored using several methods. Recognizing the trinomial squares pattern will make your work easier.
Remember, sums of squares do not factor, but sums of cubes do!
When using the sum or difference of cubes pattern, being careful with the signs.
Taking out the complete GCF in the first step will always make your work easier.
When we have factored a polynomial with four terms, most often we separated it into two groups of two terms. Remember that we can also separate it into a trinomial and then one term.
### Key Concepts
1. How to use a general strategy for factoring polynomials.
### Practice Makes Perfect
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
### 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
## Polynomial Equations
We have spent considerable time learning how to factor polynomials. We will now look at polynomial equations and solve them using factoring, if possible.
A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
We have already solved polynomial equations of degree one. Polynomial equations of degree one are linear equations are of the form
We are now going to solve polynomial equations of degree two. A polynomial equation of degree two is called a quadratic equation. 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 with (If then and we are left with no quadratic term.)
To solve quadratic equations we need methods different from the ones we used in solving linear equations. We will look at one method here and then several others in a later chapter.
### Use 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, then 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.
### Solve Quadratic Equations by Factoring
The Zero Product Property works very nicely to solve quadratic equations. The quadratic equation must be factored, with zero isolated on one side. So we must be sure to start with the quadratic equation in standard form, Then we must 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?
In the next example, the left side of the equation 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.
In the next example, when we factor the quadratic equation we will get three factors. However the first factor is a constant. We know that factor cannot equal 0.
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 greater than two by using the Zero Product Property, just like we solved quadratic equations.
### Solve Equations with Polynomial Functions
As our study of polynomial functions continues, it will often be important to know when the function will have a certain value or what points lie on the graph of the function. Our work with the Zero Product Property will be help us find these answers.
The Zero Product Property also helps us determine where the function is zero. A value of x where the function is 0, is called a zero of the function.
When the point is a point on the graph. This point is an of the graph. It is often important to know where the graph of a function crosses the axes. We will see some examples later.
### Solve Applications Modeled by Polynomial 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 polynomial 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 polynomial 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 positive results.
In some applications, negative solutions will result from the algebra, but will not be realistic for the situation.
In the next example, we will use the Pythagorean Theorem This formula gives the relation between the legs and the hypotenuse of a right triangle.
We will use this formula to in the next example.
The next example uses the function that gives the height of an object as a function of time when it is thrown from 80 feet above the ground.
### Key Concepts
1. Polynomial Equation: A polynomial equation is an equation that contains a polynomial expression. The degree of the polynomial equation is the degree of the polynomial.
2. Quadratic Equation: An equation of the form is called a quadratic equation.
3. Zero Product Property: If then either or or both.
4. How to use the Zero Product Property
5. How to solve a quadratic equation by factoring.
6. Zero of a Function: For any function f, if then x is a zero of the function.
7. How to use a problem solving strategy to solve word problems.
### 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 Equations with Polynomial Functions
In the following exercises, solve.
In the following exercises, for each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
### 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 Review Exercises
### 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.
### Factor Trinomials
Factor Trinomials of the Form
In the following exercises, factor each trinomial of the form
In the following examples, factor each trinomial of the form
Factor Trinomials of the Form
In the following exercises, factor completely using trial and error.
Factor Trinomials of the Form
In the following exercises, factor.
Factor using substitution
In the following exercises, factor using substitution.
### Factor Special Products
Factor Perfect Square Trinomials
In the following exercises, factor completely using the perfect square trinomials pattern.
Factor Differences of Squares
In the following exercises, factor completely using the difference of squares pattern, if possible.
Factor Sums and Differences of Cubes
In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.
### General Strategy for Factoring Polynomials
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
### Polynomial Equations
Use the Zero Product Property
In the following exercises, solve.
Solve Quadratic Equations by Factoring
In the following exercises, solve.
Solve Equations with Polynomial Functions
In the following exercises, solve.
In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
### Chapter Practice Test
In the following exercises, factor completely.
In the following exercises, solve |
# Rational Expressions and Functions
## Introduction
Twelve goals last season. Fifteen home runs. Nine touchdowns. Whatever the statistics, sports analysts know it. Their jobs depend on it. Compiling and analyzing sports data not only help fans appreciate their teams but also help owners and coaches decide which players to recruit, how to best use them in games, how much they should be paid, and which players to trade. Understanding this kind of data requires a knowledge of specific types of expressions and functions. In this chapter, you will work with rational expressions and perform operations on them. And you will use rational expressions and inequalities to solve real-world problems. |
# Rational Expressions and Functions
## Multiply and Divide Rational Expressions
We previously reviewed the properties of fractions and their operations. We introduced rational numbers, which are just fractions where the numerators and denominators are integers. In this chapter, we will work with fractions whose numerators and denominators are polynomials. We call this kind of expression a rational expression.
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 do the same operations with rational expressions that we did with fractions. We will simplify, add, subtract, multiply, divide and use them in applications.
### Determine the Values for Which a Rational Expression is Undefined
If the denominator is zero, the rational expression is undefined. The numerator of a rational expression may be 0—but not the denominator.
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.
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.
### Simplify Rational Expressions
A fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator. Similarly, a simplified rational expression has no common factors, other than 1, in its numerator and denominator.
For example,
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.
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.
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 have learned to factor the polynomials in the numerators and denominators in the following examples.
Every time we write a rational expression, we should make a 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.
Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. We previously introduced opposite notation: the opposite of a is and
The numerical fraction, say simplifies to . We also recognize that the numerator and denominator are opposites.
The fraction , whose numerator and denominator are opposites also simplifies to .
This tells us that is the opposite of
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. Be careful not to treat and as opposites. Recall that in addition, order doesn’t matter so . So if , then
### 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.
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
Just like we did for numerical fractions, to divide rational expressions, we multiply the first fraction by the reciprocal of the second.
Once we rewrite the division as multiplication of the first expression by the reciprocal of the second, we then factor everything and look for common factors.
Recall from Use the Language of Algebra that a complex fraction is a fraction that contains a fraction in the numerator, the denominator or both. Also, 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.
### Multiply and Divide Rational Functions
We started this section stating that a rational expression is an expression of the form where p and q are polynomials and Similarly, we define a rational function as a function of the form where and are polynomial functions and is not zero.
The domain of a rational function is all real numbers except for those values that would cause division by zero. We must eliminate any values that make
To multiply rational functions, we multiply the resulting rational expressions on the right side of the equation using the same techniques we used to multiply rational expressions.
To divide rational functions, we divide the resulting rational expressions on the right side of the equation using the same techniques we used to divide rational expressions.
### Key Concepts
1. Determine the values for which a rational expression is undefined.
2. Equivalent Fractions Property
If a, b, and c are numbers where then and
3. How to simplify a rational expression.
4. Opposites in a Rational Expression
The opposite of is An expression and its opposite divide to
5. Multiplication of Rational Expressions
If p, q, r, and s are polynomials where then
6. How to multiply rational expressions.
7. Division of Rational Expressions
If p, q, r, and s are polynomials where then
8. How to divide rational expressions.
9. How to determine the domain of a rational function.
### Practice Makes Perfect
Determine the Values for Which a Rational Expression is Undefined
In the following exercises, determine the values for which the rational expression is undefined.
Simplify Rational Expressions
In the following exercises, simplify each rational expression.
Multiply Rational Expressions
In the following exercises, multiply the rational expressions.
Divide Rational Expressions
In the following exercises, divide the rational expressions.
For the following exercises, perform the indicated operations.
Multiply and Divide Rational Functions
In the following exercises, find the domain of each function.
For the following exercises, find where and are given.
For the following exercises, find where and are given.
### 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 Functions
## Add and Subtract Rational Expressions
### Add and Subtract 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.
To add or subtract rational expressions with a common denominator, add or subtract the numerators and place the result 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.
Remember, too, we do not allow values that would make the denominator zero. What value of x should be excluded in the next example?
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. Be careful of the signs when you subtract a binomial or trinomial.
### 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.
Be careful with the signs as you work with the opposites when the fractions are being subtracted.
### 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 this example: Since the denominators are not the same, the first step was to find the least common denominator (LCD).
To find the LCD of the fractions, we factored 12 and 18 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.
When we add numerical fractions, once we found the LCD, we rewrote each fraction as an equivalent fraction with the LCD by multiplying the numerator and denominator by the same number. We are now ready to add.
We do the same thing for rational expressions. However, we leave the LCD in factored form.
Remember, we always exclude values that would make the denominator zero. What values of should we exclude in this next example?
### Add and Subtract Rational Expressions with Unlike Denominators
Now we have all the steps we need to add or subtract rational expressions with unlike denominators.
The steps used to add rational expressions are summarized here.
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.
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.
There are lots of negative signs in the next example. Be extra careful.
Things can get very messy when both fractions must be multiplied by a binomial to get the common denominator.
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.
### Add and subtract rational functions
To add or subtract rational functions, we use the same techniques we used to add or subtract polynomial functions.
### Key Concepts
1. Rational Expression Addition and Subtraction
If p, q, and r are polynomials where then
and
2. How to find the least common denominator of rational expressions.
3. How to add or subtract rational expressions.
### Practice Makes Perfect
Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, add.
In the following exercises, subtract.
Add and Subtract Rational Expressions whose Denominators are Opposites
In the following exercises, add or subtract.
Find the Least Common Denominator of Rational Expressions
In the following exercises, ⓐ find the LCD for the given rational expressions ⓑ rewrite them as equivalent rational expressions with the lowest common denominator.
Add and Subtract Rational Expressions with Unlike Denominators
In the following exercises, perform the indicated operations.
Add and Subtract Rational Functions
In the following exercises, find ⓐ ⓑ
### 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 Functions
## Simplify Complex Rational Expressions
### Simplify a Complex Rational Expression by Writing it as Division
Complex fractions are fractions in which the numerator or denominator contains a fraction. We previously 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.
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 complex rational expressions. We make sure the complex rational expression is of the form where one fraction is over one fraction. We then 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.
We follow the same procedure when the complex rational expression contains variables.
We summarize the steps here.
### 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 the 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.
We will use the same example as in . Decide which method works better for you.
Be sure to start by factoring all the denominators so you can find the LCD.
Be sure to factor the denominators first. Proceed carefully as the math can get messy!
### Key Concepts
1. How to simplify a complex rational expression by writing it as division.
2. How 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 each complex rational expression by writing it as division.
Simplify a Complex Rational Expression by Using the LCD
In the following exercises, simplify each complex rational expression by using the LCD.
In the following exercises, simplify each complex rational expression using either method.
### 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 Functions
## Solve Rational Equations
After defining the terms ‘expression’ and ‘equation’ earlier, 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 a rational equation.
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.
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 to a rational equation.
We note any possible extraneous solutions, c, by writing next to the equation.
The steps of this method are shown.
We always start by noting the values that would cause any denominators to be zero.
In the next example, the last denominators is a difference of squares. Remember to factor it first to find the LCD.
In the next example, the first denominator is a trinomial. Remember to factor it first to find the LCD.
The equation we solved in the previous example had only one algebraic solution, but it was an extraneous solution. That left us with no solution to the equation. In the next example we get two algebraic solutions. Here one or both could be extraneous solutions.
In some cases, all the algebraic solutions are extraneous.
### Use Rational Functions
Working with functions that are defined by rational expressions often lead to rational equations. Again, we use the same techniques to solve them.
### 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.
When we developed the point-slope formula from our slope formula, we cleared the fractions by multiplying by the LCD.
In the next example, we will use the same technique with the formula for slope that we used to get the point-slope form of an equation of a line through the point We will add one more step to solve for y.
Remember to multiply both sides by the LCD in the next example.
### Key Concepts
1. How to solve equations with rational expressions.
### Practice Makes Perfect
Solve Rational Equations
In the following exercises, solve each rational equation.
Solve Rational Equations that Involve Functions
Solve a Rational Equation for a Specific Variable
In the following exercises, solve.
### 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 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 Functions
## Solve Applications with Rational Equations
### 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.”
Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved rational equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.
Notice in the last example that when we cleared the fractions by multiplying by the LCD, the result is the same as if we had cross-multiplied.
For any proportion, we get the same result when we clear the fractions by multiplying by the LCD as when we cross-multiply.
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 each other and the units in the denominators must also match each other.
### 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 have the same ratio.
For example, the two triangles in are similar. Each side of is four 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.
On the map, Seattle, Portland, and Boise form a triangle. The distance between the cities is measured in inches. The figure on the left below represents the triangle formed by the cities on the map. The actual distance from Seattle to Boise is 400 miles.
We can use similar figures to find heights that we cannot directly measure.
### 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 the wind affects the speed of a plane. 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
### Solve Work Applications
The weekly gossip magazine has a big story about the Princess’ baby and the editor wants the magazine to be printed as soon as possible. She has asked the printer to run an extra printing press to get the printing done more quickly. Press #1 takes 6 hours to do the job and Press #2 takes 12 hours to do the job. How long will it take the printer to get the magazine printed with both presses running together?
This is a typical ‘work’ application. There are three quantities involved here—the time it would take each of the two presses to do the job alone and the time it would take for them to do the job together.
If Press #1 can complete the job in 6 hours, in one hour it would complete of the job.
If Press #2 can complete the job in 12 hours, in one hour it would complete of the job.
We will let t be the number of hours it would take the presses to print the magazines with both presses running together. So in 1 hour working together they have completed 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 the presses to complete the job if they worked together, we solve for
A chart will help us organize the information. We are looking for how many hours it would take to complete the job with both presses running together.
Keep in mind, it should take less time for two presses to complete a job working together than for either press to do it alone.
### Solve Direct Variation Problems
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.
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 here.
Now we’ll solve an application of direct variation.
### 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’.
### Key Concepts
1. A proportion is an equation of the form where The proportion is read “a is to b as c is to d.”
2. Property of Similar Triangles
If is similar to then their corresponding angle measure are equal and their corresponding sides have the same ratio.
3. Direct Variation
4. Inverse Variation
### Practice Makes Perfect
Solve Proportions
In the following exercises, solve each proportion.
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, the triangles are similar. Find the length of the indicated side.
In the following exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle. The actual distance from New York to Chicago is 800 miles.
In the following exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle. The actual distance from Atlanta to New Orleans is 420 miles.
In the following exercises, answer each question.
Solve Uniform Motion Applications
In the following exercises, solve the application problem provided.
Solve Work Applications
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
In the following exercises, write an inverse variation equation to solve the following problems.
### 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 Functions
## Solve Rational Inequalities
### Solve Rational Inequalities
We learned to solve linear inequalities after learning to solve linear equations. The techniques were very much the same with one major exception. When we multiplied or divided by a negative number, the inequality sign reversed.
Having just learned to solve rational equations we are now ready to solve rational inequalities. A rational inequality is an inequality that contains a rational expression.
Inequalities such as and are rational inequalities as they each contain a rational expression.
When we solve a rational inequality, we will use many of the techniques we used solving linear inequalities. We especially must remember that when we multiply or divide by a negative number, the inequality sign must reverse.
Another difference is that we must carefully consider what value might make the rational expression undefined and so must be excluded.
When we solve an equation and the result is we know there is one solution, which is 3.
When we solve an inequality and the result is we know there are many solutions. We graph the result to better help show all the solutions, and we start with 3. Three becomes a critical point and then we decide whether to shade to the left or right of it. The numbers to the right of 3 are larger than 3, so we shade to the right.
To solve a rational inequality, we first must write the inequality with only one quotient on the left and 0 on the right.
Next we determine the critical points to use to divide the number line into intervals. A critical point is a number which make the rational expression zero or undefined.
We then will evaluate the factors of the numerator and denominator, and find the quotient in each interval. This will identify the interval, or intervals, that contains all the solutions of the rational inequality.
We write the solution in interval notation being careful to determine whether the endpoints are included.
We summarize the steps for easy reference.
The next example requires that we first get the rational inequality into the correct form.
In the next example, the numerator is always positive, so the sign of the rational expression depends on the sign of the denominator.
The next example requires some work to get it into the needed form.
### Solve an Inequality with Rational Functions
When working with rational functions, it is sometimes useful to know when the function is greater than or less than a particular value. This leads to a rational inequality.
In economics, the function is used to represent the cost of producing x units of a commodity. The average cost per unit can be found by dividing by the number of items Then, the average cost per unit is
### Key Concepts
1. Solve a rational inequality.
### Section Exercises
### Practice Makes Perfect
Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
Solve an Inequality with Rational Functions
In the following exercises, solve each rational function inequality and write the solution in interval notation.
### 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 Review Exercises
### Simplify, Multiply, and Divide 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.
Simplify Rational Expressions
In the following exercises, simplify.
Multiply Rational Expressions
In the following exercises, multiply.
Divide Rational Expressions
In the following exercises, divide.
Multiply and Divide Rational Functions
### Add and Subtract Rational Expressions
Add and Subtract Rational Expressions with a Common Denominator
In the following exercises, perform the indicated operations.
Add and Subtract Rational Expressions Whose Denominators Are Opposites
In the following exercises, add and subtract.
Find the Least Common Denominator of Rational Expressions
In the following exercises, find the LCD.
Add and Subtract Rational Expressions with Unlike Denominators
In the following exercises, perform the indicated operations.
Add and Subtract Rational Functions
In the following exercises, find where and are given.
In the following exercises, find where and are given.
### 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.
### 7.4 Solve Rational Equations
Solve Rational Equations
In the following exercises, solve.
Solve Rational Equations that Involve Functions
Solve a Rational Equation for a Specific Variable
In the following exercises, solve for the indicated variable.
### Solve Applications with Rational Equations
Solve Proportions
In the following exercises, solve.
Solve Using Proportions
In the following exercises, solve.
Solve Similar Figure Applications
In the following exercises, solve.
Solve Uniform Motion Applications
In the following exercises, solve.
Solve Work Applications
In the following exercises, solve.
Solve Direct Variation Problems
In the following exercises, solve.
Solve Inverse Variation Problems
In the following exercises, solve.
### Solve Rational Inequalities
Solve Rational Inequalities
In the following exercises, solve each rational inequality and write the solution in interval notation.
Solve an Inequality with Rational Functions
In the following exercises, solve each rational function inequality and write the solution in interval notation
### 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 each rational inequality and write the solution in interval notation.
In the following exercises, find given and
In the following exercises, solve. |
# Roots and Radicals
## Introduction
Imagine charging your cell phone is less than five seconds. Consider cleaning radioactive waste from contaminated water. Think about filtering salt from ocean water to make an endless supply of drinking water. Ponder the idea of bionic devices that can repair spinal injuries. These are just of few of the many possible uses of a material called graphene. Materials scientists are developing a material made up of a single layer of carbon atoms that is stronger than any other material, completely flexible, and conducts electricity better than most metals. Research into this type of material requires a solid background in mathematics, including understanding roots and radicals. In this chapter, you will learn to simplify expressions containing roots and radicals, perform operations on radical expressions and equations, and evaluate radical functions. |
# Roots and Radicals
## Simplify Expressions with Roots
### Simplify Expressions with Roots
In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,
Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.
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? We use a radical sign, and write, which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are grouping symbols.
We also use the radical sign for the square root of zero. Because Notice that zero has only one square root.
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,
Can we simplify Is there a number whose square is
Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to The square root of a negative number is not a real number.
So far we have only talked about squares and square roots. Let’s 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 −5 to 5. See .
Notice the signs in the table. 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 −2 to help you see this.
We will now extend the square root definition to higher roots.
Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for
We can refer to to help find higher roots.
Could we have an even root of a negative number? 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.
We will apply these properties in the next two examples.
In this example be alert for the negative signs as well as even and odd powers.
### Estimate and Approximate Roots
When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the what is the value of or In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.
To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.
Similarly, to estimate we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.
There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the key on your calculator. To find a cube root, or any root with higher index, you will use the key.
When you use these keys, you get an approximate value. 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
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. The fourth powers are close to 93, but not equal to 93.
### Simplify Variable Expressions with Roots
The odd root of a number can be either positive or negative. For example,
But what about an even root? We want the principal root, so
But notice,
How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value. So we say that when n is even This guarantees the principal root is positive.
What about square roots of higher powers of variables? The Power Property of Exponents says So if we square a, the exponent will become 2m.
Looking now at the square root,
We apply this concept in the next example.
The next example uses the same idea for highter roots.
In the next example, we now have a coefficient in front of the variable. The concept works in much the same way.
But notice and no absolute value sign is needed as u4 is always positive.
This example just takes the idea farther as it has roots of higher index.
The next examples have two variables.
### Key Concepts
1. Square Root Notation
2.
3. Properties of
4. Simplifying Odd and Even Roots
### Practice Makes Perfect
Simplify Expressions with Roots
In the following exercises, simplify.
Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
In the following exercises, approximate each root and round to two decimal places.
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.
### 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. |
# Roots and Radicals
## Simplify Radical Expressions
### Use the Product Property to Simplify Radical Expressions
We will simplify radical expressions in a way similar to how we simplified fractions. A fraction is simplified if there are no common factors in the numerator and denominator. To simplify a fraction, we look for any common factors in the numerator and denominator.
A radical expression, is considered simplified if it has no factors of So, to simplify a radical expression, we look for any factors in the radicand that are powers of the index.
For example, is considered simplified because there are no perfect square factors in 5. But is not simplified because 12 has a perfect square factor of 4.
Similarly, is simplified because there are no perfect cube factors in 4. But is not simplified because 24 has a perfect cube factor of 8.
To simplify radical expressions, we will also use some properties of roots. The properties we will use to simplify radical expressions are similar to the properties of exponents. We know that The corresponding of Product Property of Roots says that
We use the Product Property of Roots to remove all perfect square factors from a square root.
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.
Be careful to write your integer so that it is not confused with the index. The expression is very different from
We will apply this method in the next example. It may be helpful to have a table of perfect squares, cubes, and fourth powers.
The next example is much like the previous examples, but with variables. Don’t forget to use the absolute value signs when taking an even root of an expression with a variable in the radical.
We follow the same procedure when there is a coefficient in the radicand. In the next example, both the constant and the variable have perfect square factors.
In the next example, we continue to use the same methods even though there are more than one variable under the radical.
We have seen how to use the order of operations to simplify some expressions with radicals. 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 since one term contains a radical and the other does not. The next example also 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 Radical Expressions
Whenever you have to simplify a radical expression, the first step you should take is to determine whether the radicand is a perfect power of the index. If not, check the numerator and denominator for any common factors, and remove them. You may find a fraction in which both the numerator and the denominator are perfect powers of the index.
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 root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect power of the index, we simplify the numerator and denominator separately.
Be sure to simplify the fraction in the radicand first, if possible.
In the next example, there is nothing to simplify in the denominators. Since the index on the radicals is the same, we can use the Quotient Property again, to combine them into one radical. We will then look to see if we can simplify the expression.
### Key Concepts
1. Simplified Radical Expression
2. Product Property of n
3. How to simplify a radical expression using the Product Property
4. Quotient Property of Radical Expressions
5. How to simplify a radical expression using the Quotient Property.
### Practice Makes Perfect
Use the Product Property to Simplify Radical Expressions
In the following exercises, use the Product Property to simplify radical expressions.
In the following exercises, simplify using absolute value signs as needed.
Use the Quotient Property to Simplify Radical Expressions
In the following exercises, use the Quotient Property to simplify square roots.
### 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
## Simplify 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.
So But we know also Then it must be that
This same logic can be used for any positive integer exponent n to show that
The denominator of the rational exponent is the index of the radical.
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, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.
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
We can look at in two ways. Remember the Power Property tells us to multiply the exponents and so and both equal If we write these expressions in radical form, we get
This leads us to the following definition.
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, before raising it to the power indicated.
Remember that The negative sign in the exponent does not change the sign of the expression.
### Use the Properties of Exponents to Simplify Expressions with Rational Exponents
The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponenets here to have them for reference as we simplify expressions.
We will apply these properties in the next example.
Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.
We will use both the Product Property and the Quotient Property in the next example.
### Key Concepts
1. Rational Exponent
2. Rational Exponent
3. Properties of Exponents
### 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. Assume all variables are positive.
### 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
## Add, Subtract, and Multiply Radical Expressions
### Add and Subtract Radical Expressions
Adding radical expressions with the same index and the same radicand is just like adding like terms. We call radicals with the same index and the same radicand like radicals to remind us they work the same as like terms.
We add and subtract like radicals in the same way we add and subtract like terms. We know that is Similarly we add and the result is
Think about adding like terms with variables as you do the next few examples. When you have like radicals, you just add or subtract the coefficients. When the radicals are not like, you cannot combine the terms.
For radicals to be like, they must have the same index and radicand. When the radicands contain more than one variable, as long as all the variables and their exponents are identical, the radicands are the same.
Remember that we always simplify radicals by removing the largest factor from the radicand that is a power of the index. Once each radical is simplified, we can then decide if they are like radicals.
In the next example, we will remove both constant and variable factors from the radicals. Now that we have practiced taking both the even and odd roots of variables, it is common practice at this point for us to assume all variables are greater than or equal to zero so that absolute values are not needed. We will use this assumption throughout the rest of this chapter.
### Multiply Radical Expressions
We have used the Product Property of Roots to simplify square roots by removing the perfect square factors. We can use the Product Property of 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 Roots so we see both ways together.
When we multiply two radicals they must have the same index. Once we multiply the radicals, we then look for factors that are a power of the index 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 12xy. Keep this in mind as you do these examples.
We follow the same procedures when there are variables in the radicands.
### Use Polynomial Multiplication to Multiply Radical Expressions
In the next a few examples, we will use the Distributive Property to multiply expressions with radicals. First we will distribute and then simplify the radicals 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.
Recognizing some special products made our work easier when we multiplied binomials earlier. This is true when we multiply radicals, too. The special product formulas we used are shown here.
We will use the special product formulas in the next few examples. We will start with the Product of Binomial Squares Pattern.
In the next example, we will use the Product of Conjugates Pattern. Notice that the final product has no radical.
### Key Concepts
1. Product Property of Roots
2. Special Products
### Practice Makes Perfect
Add and Subtract Radical Expressions
In the following exercises, simplify.
Multiply Radical Expressions
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Radical Expressions
In the following exercises, multiply.
Mixed Practice
### 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 Radical Expressions
### Divide Radical Expressions
We have used the Quotient Property of Radical Expressions to simplify roots of fractions. We will need to use this property ‘in reverse’ to simplify a fraction with radicals.
We give the Quotient Property of Radical Expressions again for easy reference. Remember, we assume all variables are greater than or equal to zero so that no absolute value bars are needed.
We will use the Quotient Property of Radical Expressions when the fraction we start with is the quotient of two radicals, and neither radicand is a perfect power of the index. When we write the fraction in a single radical, we may find common factors in the numerator and denominator.
### Rationalize a One Term Denominator
Before the calculator became a tool of everyday life, approximating the value of a fraction with a radical in the denominator 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. 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.
This process is still used today, and is useful in other areas of mathematics, too.
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 radical.
Similarly, a radical expression is not considered simplified if the radicand contains a fraction.
To rationalize a denominator with a square root, 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.
When we rationalized a square root, we multiplied the numerator and denominator by a square root that would give us a perfect square under the radical in the denominator. When we took the square root, the denominator no longer had a radical.
We will follow a similar process to rationalize higher roots. To rationalize a denominator with a higher index radical, we multiply the numerator and denominator by a radical that would give us a radicand that is a perfect power of the index. When we simplify the new radical, the denominator will no longer have a radical.
For example,
We will use this technique in the next examples.
### 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.
Notice we did not distribute the 5 in the answer of the last example. By leaving the result factored we can see if there are any factors that may be common to both the numerator and denominator.
Be careful of the signs when multiplying. The numerator and denominator look very similar when you multiply by the conjugate.
### Key Concepts
1. Quotient Property of Radical Expressions
2. Simplified Radical Expressions
### Practice Makes Perfect
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, simplify.
### 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 Radical Equations
### Solve Radical Equations
In this section we will solve equations that have a variable in the radicand of a radical expression. An equation of this type is called a radical equation.
As usual, when solving these equations, what we do to one side of an equation we must do to the other side as well. Once we isolate the radical, our strategy will be to raise both sides of the equation to the power of the index. This will eliminate the radical.
Solving radical equations containing an even index by raising both sides to the power of the index may introduce an algebraic solution that would not be a solution to the original radical equation. Again, we call this an extraneous solution as we did when we solved rational equations.
In the next example, we will see how to solve a radical equation. Our strategy is based on raising a radical with index n to the nth power. This will eliminate the radical.
When we use a radical sign, it indicates the principal or positive root. If an equation has a radical with an even index equal to a negative number, that equation will have no solution.
If one side of an equation with a square root is a binomial, we use the Product of Binomial Squares Pattern when we square it.
Don’t forget the middle term!
When the index of the radical is 3, we cube both sides to remove the radical.
Sometimes an equation will contain rational exponents instead of a radical. We use the same techniques to solve the equation as when we have a radical. We raise each side of the equation to the power of the denominator of the rational exponent. Since we have for example,
Remember, and
Sometimes the solution of a radical equation results in two algebraic solutions, but one of them may be an extraneous solution!
When there is a coefficient in front of the radical, we must raise it to the power of the index, too.
### Solve Radical Equations with Two Radicals
If the radical equation has two radicals, we start out by isolating one of them. It often works out easiest to isolate the more complicated radical first.
In the next example, when one radical is isolated, the second radical is also isolated.
Sometimes after raising both sides of an equation to a power, 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 raise both sides of the equation to the power of the index again.
We summarize the steps here. We have adjusted our previous steps to include more than one radical in the equation This procedure will now work for any radical equations.
Be careful as you square binomials in the next example. Remember the pattern is or
### Use Radicals in Applications
As you progress through your college courses, you’ll encounter formulas that include radicals in many disciplines. We will modify our Problem Solving Strategy for Geometry Applications slightly to give us a plan for solving applications with formulas from any discipline.
One application of radicals has to do with the effect of gravity on falling objects. The formula allows us to determine how long it will take a fallen object to hit the gound.
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. Binomial Squares
2. Solve a Radical Equation
3. Problem Solving Strategy for Applications with Formulas
4. Falling Objects
5. Skid Marks and Speed of a Car
### Practice Makes Perfect
Solve Radical Equations
In the following exercises, solve.
Solve Radical Equations with Two Radicals
In the following exercises, solve.
Use Radicals 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
## Use Radicals in Functions
### Evaluate a Radical Function
In this section we will extend our previous work with functions to include radicals. If a function is defined by a radical expression, we call it a radical function.
The square root function is
The cube root function is
To evaluate a radical function, we find the value of f(x) for a given value of x just as we did in our previous work with functions.
We follow the same procedure to evaluate cube roots.
The next example has fourth roots.
### Find the Domain of a Radical Function
To find the domain and range of radical functions, we use our properties of radicals. For a radical with an even index, we said the radicand had to be greater than or equal to zero as even roots of negative numbers are not real numbers. For an odd index, the radicand can be any real number. We restate the properties here for reference.
So, to find the domain of a radical function with even index, we set the radicand to be greater than or equal to zero. For an odd index radical, the radicand can be any real number.
The next example involves a cube root and so will require different thinking.
### Graph Radical Functions
Before we graph any radical function, we first find the domain of the function. For the function, the index is even, and so the radicand must be greater than or equal to 0.
This tells us the domain is and we write this in interval notation as
Previously we used point plotting to graph the function, We chose x-values, substituted them in and then created a chart. Notice we chose points that are perfect squares in order to make taking the square root easier.
Once we see the graph, we can find the range of the function. The y-values of the function are greater than or equal to zero. The range then is
In our previous work graphing functions, we graphed but we did not graph the function We will do this now in the next example.
### Key Concepts
1. Properties of
2. Domain of a Radical Function
### Practice Makes Perfect
Evaluate a Radical Function
In the following exercises, evaluate each function.
Find the Domain of a Radical Function
In the following exercises, find the domain of the function and write the domain in interval notation.
Graph Radical Functions
In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.
### 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
## Use the Complex Number System
### Evaluate the Square Root of a Negative Number
Whenever we have a situation where we have a square root of a negative number we say there is no real number that equals that square root. For example, to simplify we are looking for a real number x so that x2 = –1. Since all real numbers squared are positive numbers, there is no real number that equals –1 when squared.
Mathematicians have often expanded their numbers systems as needed. They added 0 to the counting numbers to get the whole numbers. When they needed negative balances, they added negative numbers to get the integers. When they needed the idea of parts of a whole they added fractions and got the rational numbers. Adding the irrational numbers allowed numbers like All of these together gave us the real numbers and so far in your study of mathematics, that has been sufficient.
But now we will expand the real numbers to include the square roots of negative numbers. We start by defining the imaginary unit as the number whose square is –1.
We will use the imaginary unit to simplify the square roots of negative numbers.
We will use this definition in the next example. Be careful that it is clear that the i is not under the radical. Sometimes you will see this written as to emphasize the i is not under the radical. But the is considered standard form.
Now that we are familiar with the imaginary number i, we can expand our concept of the number system to include imaginary numbers. The complex number system includes the real numbers and the imaginary numbers. A complex number is of the form a + bi, where a, b are real numbers. We call a the real part and b the imaginary part.
A complex number is in standard form when written as where a and b are real numbers.
If then becomes and is a real number.
If then is an imaginary number.
If then becomes and is called a pure imaginary number.
We summarize this here.
The standard form of a complex number is so this explains why the preferred form is when
The diagram helps us visualize the complex number system. It is made up of both the real numbers and the imaginary numbers.
### Add or Subtract Complex Numbers
We are now ready to perform the operations of addition, subtraction, multiplication and division on the complex numbers—just as we did with the real numbers.
Adding and subtracting complex numbers is much like adding or subtracting like terms. We add or subtract the real parts and then add or subtract the imaginary parts. Our final result should be in standard form.
Remember to add both the real parts and the imaginary parts in this next example.
### Multiply Complex Numbers
Multiplying complex numbers is also much like multiplying expressions with coefficients and variables. There is only one special case we need to consider. We will look at that after we practice in the next two examples.
In the next example, we multiply the binomials using the Distributive Property or FOIL.
In the next example, we could use FOIL or the Product of Binomial Squares Pattern.
Since the square root of a negative number is not a real number, when we have the square roots of two negative numbers, we cannot use the Product Property for Radicals. In order to multiply square roots of negative numbers we should first write them as complex numbers, using This is one place students tend to make errors, so be careful when you see multiplying with a negative square root.
In the next example, each binomial has a square root of a negative number. Before multiplying, each square root of a negative number must be written as a complex number.
We first looked at conjugate pairs when we studied polynomials. We said that 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 is called a conjugate pair and is of the form
A complex conjugate pair is very similar. For a complex number of the form its conjugate is Notice they have the same first term and the same last term, but one is a sum and one is a difference.
We will multiply a complex conjugate pair in the next example.
From our study of polynomials, we know the product of conjugates is always of the form The result is called a difference of squares. We can multiply a complex conjugate pair using this pattern.
The last example we used FOIL. Now we will use the Product of Conjugates Pattern.
Notice this is the same result we found in .
When we multiply complex conjugates, the product of the last terms will always have an which simplifies to
This leads us to the Product of Complex Conjugates Pattern:
### Divide Complex Numbers
Dividing complex numbers is much like rationalizing a denominator. We want our result to be in standard form with no imaginary numbers in the denominator.
We summarize the steps here.
Be careful as you find the conjugate of the denominator.
### Simplify Powers of i
The powers of make an interesting pattern that will help us simplify higher powers of i. Let’s evaluate the powers of to see the pattern.
We summarize this now.
If we continued, the pattern would keep repeating in blocks of four. We can use this pattern to help us simplify powers of i. Since i4 = 1, we rewrite each power, i, as a product using i4 to a power and another power of i.
We rewrite it in the form where the exponent, q, is the quotient of n divided by 4 and the exponent, r, is the remainder from this division. For example, to simplify i57, we divide 57 by 4 and we get 14 with a remainder of 1. In other words, So we write and then simplify from there.
### Key Concepts
1. Square Root of a Negative Number
2. Product of Complex Conjugates
3. How to Divide Complex Numbers
### Section Exercises
### Practice Makes Perfect
Evaluate the Square Root of a Negative Number
In the following exercises, write each expression in terms of i and simplify if possible.
Add or Subtract Complex Numbers In the following exercises, add or subtract.
Multiply Complex Numbers
In the following exercises, multiply.
In the following exercises, multiply using the Product of Binomial Squares Pattern.
In the following exercises, multiply.
In the following exercises, multiply using the Product of Complex Conjugates Pattern.
Divide Complex Numbers
In the following exercises, divide.
Simplify Powers of
In the following exercises, simplify.
### 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 how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
### Chapter Review Exercises
### Simplify Expressions with Roots
Simplify Expressions with Roots
In the following exercises, simplify.
Estimate and Approximate Roots
In the following exercises, estimate each root between two consecutive whole numbers.
In the following exercises, approximate each root and round to two decimal places.
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.
### Simplify Radical Expressions
Use the Product Property to Simplify Radical Expressions
In the following exercises, use the Product Property to simplify radical expressions.
In the following exercises, simplify using absolute value signs as needed.
Use the Quotient Property to Simplify Radical Expressions
In the following exercises, use the Quotient Property to simplify square roots.
### 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.
### Add, Subtract and Multiply Radical Expressions
Add and Subtract Radical Expressions
In the following exercises, simplify.
Multiply Radical Expressions
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Radical Expressions
In the following exercises, multiply.
### Divide Radical Expressions
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, simplify.
### Solve Radical Equations
Solve Radical Equations
In the following exercises, solve.
Solve Radical Equations with Two Radicals
In the following exercises, solve.
Use Radicals in Applications
In the following exercises, solve. Round approximations to one decimal place.
### Use Radicals in Functions
Evaluate a Radical Function
In the following exercises, evaluate each function.
Find the Domain of a Radical Function
In the following exercises, find the domain of the function and write the domain in interval notation.
Graph Radical Functions
In the following exercises, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range.
### Use the Complex Number System
Evaluate the Square Root of a Negative Number
In the following exercises, write each expression in terms of i and simplify if possible.
Add or Subtract Complex Numbers
In the following exercises, add or subtract.
Multiply Complex Numbers
In the following exercises, multiply.
In the following exercises, multiply using the Product of Binomial Squares Pattern.
In the following exercises, multiply using the Product of Complex Conjugates Pattern.
Divide Complex Numbers
In the following exercises, divide.
Simplify Powers of
In the following exercises, simplify.
### Practice Test
In the following exercises, simplify using absolute values as necessary.
In the following exercises, simplify. Assume all variables are positive.
In the following exercises, solve.
In the following exercise, ⓐ find the domain of the function ⓑ graph the function ⓒ use the graph to determine the range. |
# Quadratic Equations and Functions
## Introduction
Blink your eyes. You’ve taken a photo. That’s what will happen if you are wearing a contact lens with a built-in camera. Some of the same technology used to help doctors see inside the eye may someday be used to make cameras and other devices. These technologies are being developed by biomedical engineers using many mathematical principles, including an understanding of quadratic equations and functions. In this chapter, you will explore these kinds of equations and learn to solve them in different ways. Then you will solve applications modeled by quadratics, graph them, and extend your understanding to quadratic inequalities. |
# Quadratic Equations and Functions
## Solve Quadratic Equations Using the Square Root Property
A quadratic equation is an equation of the form ax2 + bx + c = 0, where . Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2. 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 learn three other methods to use in case a quadratic equation cannot be factored.
### Solve Quadratic Equations of the form 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 x2 = 9.
We can easily use factoring to find the solutions of similar equations, like x2 = 16 and x2 = 25, because 16 and 25 are perfect squares. In each case, we would get two solutions, and
But what happens when we have an equation like x2 = 7? Since 7 is not a perfect square, we cannot solve the equation by factoring.
Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. Also, (−13)2 = 169, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169. So, every positive number has two square roots—one positive and one negative. We earlier defined the square root of a number in this way:
Since these equations are all of the form x2 = k, the square root definition tells us the solutions are the two square roots of k. This leads to the Square Root Property.
Notice that the Square Root Property gives two solutions to an equation of the form x2 = k, the principal square root of and its opposite. We could also write the solution as We read this as x equals positive or negative the square root of k.
Now we will solve the equation x2 = 9 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 x2 = 7.
We cannot simplify , so we leave the answer as a radical.
The steps to take to use the Square Root Property to solve a quadratic equation are listed here.
In order to use the Square Root Property, the coefficient of the variable term must equal one. In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property.
The Square Root Property states ‘If ,’ What will happen if This will be the case in the next example.
Our method also works when fractions occur in the equation; we solve as any equation with fractions. In the next example, 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 of the form a(x − h)2 = k as well. Notice that the quadratic term, x, in the original form ax2 = k is replaced with (x − h).
The first step, like before, is to isolate the term that has the variable squared. In this case, a binomial is being squared. Once the binomial is isolated, by dividing each side by the coefficient of a, then the Square Root Property can be used on (x − h)2.
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 term.
Sometimes the solutions are complex numbers.
The left sides of the equations in the next two examples do not seem to be of the form a(x − h)2. But they are perfect square trinomials, so we will factor to put them in the form we need.
### Key Concepts
1. Square Root Property
How to solve a quadratic equation using the square root property.
### Practice Makes Perfect
Solve Quadratic Equations of the Form
In the following exercises, solve each equation.
Solve Quadratic Equations of the Form
In the following exercises, solve each equation.
### Mixed Practice
In the following exercises, solve using the Square Root Property.
### Writing Exercises
### Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
Choose how would you respond to the statement “I can solve quadratic equations of the form a times the square of x minus h equals k using the Square Root Property.” “Confidently,” “with some help,” or “No, I don’t get it.”
ⓑ 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 and Functions
## 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, which is important for our work on conics later.
### Complete the Square of a Binomial Expression
In the last section, we were able to use the Square Root Property to solve the equation (y − 7)2 = 12 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 look at two examples to help us recognize the patterns.
We restate the patterns here for reference.
We can use this pattern to “make” a perfect square.
We will start with the expression x2 + 6x. Since there is a plus sign between the two terms, we will use the (a + b)2 pattern, a2 + 2ab + b2 = (a + b)2.
We ultimately need to find the last term of this trinomial that will make it a perfect square trinomial. To do that we will need to find b. But first we start with determining a. Notice that the first term of x2 + 6x is a square, x2. This tells us that a = x.
What number, b, when multiplied with 2x gives 6x? It would have to be 3, which is So b = 3.
Now to complete the perfect square trinomial, we will find the last term by squaring b, which is 32 = 9.
We can now factor.
So we found that adding 9 to x2 + 6x ‘completes the square’, and we write it as (x + 3)2.
### 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 x2 + 6x = 40, and we want to complete the square on the left, we will add 9 to both sides of the equation.
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.
The steps to solve a quadratic equation by completing the square are listed here.
When we solve an equation by completing the square, the answers will not always be integers.
In the previous example, our solutions were complex numbers. In the next example, the solutions will be irrational numbers.
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 on 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 since it says “If then a = 0 or b = 0.” Instead, we multiply the factors and then put the equation into 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 coefficient of x2 is 1, so the left side of the equation is of the form x2 + bx + c. If the x2 term has a coefficient other than 1, we take some preliminary steps to make the coefficient equal to 1.
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 coefficient of the x2 must be 1. 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.
Now that we have seen that the coefficient of x2 must be 1 for us to complete the square, we update our procedure for solving a quadratic equation by completing the square to include equations of the form ax2 + bx + c = 0.
### Key Concepts
1. Binomial Squares Pattern
If a and b are real numbers,
2. How to Complete a Square
3. How to solve a quadratic equation of the form ax2 + bx + c = 0 by completing the square.
### 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
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the form
In the following exercises, solve by completing the square.
### 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 and Functions
## Solve Quadratic Equations Using the Quadratic Formula
### 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’. Mathematicians look for patterns when they do things over and over in order to make their work easier. 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 using the general form of a quadratic equation to solve a quadratic equation for x.
We start with the standard form of a quadratic equation and solve it for x by completing the square.
To use the Quadratic Formula, we substitute the values of a, b, and c from the standard form into the expression on the right side of the formula. Then we simplify the expression. The result is the pair of solutions to the quadratic equation.
Notice the formula is an equation. Make sure you use both sides of the 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 “x =”.
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.
When we substitute a, b, and c into the Quadratic Formula and the radicand is negative, the quadratic equation will have imaginary or complex solutions. We will see this in the next example.
Remember, to use the Quadratic Formula, the equation must be written in standard form, ax2 + bx + c = 0. 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 (x − 3)2 = 0. We know from the Zero Product Property that this equation has only one solution,x = 3.
We will see in the next example how using the Quadratic Formula to solve an equation whose standard form is a perfect square trinomial equal to 0 gives just one solution. Notice that once the radicand is simplified it becomes 0 , which leads to only one solution.
### Use the Discriminant to Predict the Number and Type of Solutions of a Quadratic Equation
When we solved the quadratic equations in the previous examples, sometimes we got two real solutions, one real solution, and sometimes two complex solutions. Is there a way to predict the number and type of solutions to a quadratic equation without actually solving the equation?
Yes, the expression under the radical of the Quadratic Formula makes it easy for us to determine the number and type of solutions. This expression is called the discriminant.
Let’s look at the discriminant of the equations in some of the examples and the number and type of solutions to those quadratic equations.
### Identify the Most Appropriate Method to Use to Solve a Quadratic Equation
We summarize the four methods that we have used to solve quadratic equations below.
Given that we have four methods to use to solve a quadratic equation, how do you decide which one to use? Factoring is often the quickest method and so we try it first. If the equation is or we use the Square Root Property. For any other equation, it is probably best to use the Quadratic Formula. Remember, you can solve any quadratic equation by using the Quadratic Formula, but that is not always the easiest method.
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 the list of methods 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.
The next example uses this strategy to decide how to solve each quadratic equation.
### Key Concepts
1. Quadratic Formula
2. How to solve a quadratic equation using the Quadratic Formula.
3. Using the Discriminant, b2 − 4ac, to Determine the Number and Type of Solutions of a Quadratic Equation
4. Methods to Solve Quadratic Equations:
5. How 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 Real Solutions of a Quadratic Equation
In the following exercises, determine the number of real solutions for 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.
### 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? |
# Quadratic Equations and Functions
## Solve Equations in Quadratic Form
### Solve Equations in Quadratic Form
Sometimes when we factored trinomials, the trinomial did not appear to be in the ax2 + bx + c form. So we factored by substitution allowing us to make it fit the ax2 + bx + c form. We used the standard for the substitution.
To factor the expression x4 − 4x2 − 5, we noticed the variable part of the middle term is x2 and its square, x4, is the variable part of the first term. (We know ) So we let u = x2 and factored.
Similarly, sometimes an equation is not in the ax2 + bx + c = 0 form but looks much like a quadratic equation. Then, we can often make a thoughtful substitution that will allow us to make it fit the ax2 + bx + c = 0 form. If we can make it fit the form, we can then use all of our methods to solve quadratic equations.
Notice that in the quadratic equation ax2 + bx + c = 0, the middle term has a variable, x, and its square, x2, is the variable part of the first term. Look for this relationship as you try to find a substitution.
Again, we will use the standard u to make a substitution that will put the equation in quadratic form. If the substitution gives us an equation of the form ax2 + bx + c = 0, we say the original equation was of quadratic form.
The next example shows the steps for solving an equation in quadratic form.
We summarize the steps to solve an equation in quadratic form.
In the next example, the binomial in the middle term, (x − 2) is squared in the first term. If we let u = x − 2 and substitute, our trinomial will be in ax2 + bx + c form.
In the next example, we notice that Also, remember that when we square both sides of an equation, we may introduce extraneous roots. Be sure to check your answers!
Substitutions for rational exponents can also help us solve an equation in quadratic form. Think of the properties of exponents as you begin the next example.
In the next example, we need to keep in mind the definition of a negative exponent as well as the properties of exponents.
### Key Concepts
1. How to solve equations in quadratic form.
### Practice Makes Perfect
Solve Equations in Quadratic Form
In the following exercises, solve.
### 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? |
# Quadratic Equations and Functions
## Solve Applications of Quadratic Equations
### Solve Applications Modeled by Quadratic Equations
We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications.
Let’s first summarize the methods we now have to solve quadratic equations.
As you solve each equation, choose the method that is most convenient for you to work the problem. As a reminder, we will copy our usual Problem-Solving Strategy here so we can follow the steps.
We have solved number applications that involved consecutive even and odd integers, by modeling the situation with linear equations. Remember, we noticed each even integer is 2 more than the number preceding it. If we call the first one n, then the next one is n + 2. The next one would be n + 2 + 2 or n + 4. This is also true when we use odd integers. One set of even integers and one set of odd integers are shown below.
Some applications of odd or even consecutive integers are modeled by quadratic equations. The notation above will be helpful as you name the variables.
We will use the formula for the area of a triangle to solve the next example.
Recall that when we solve geometric applications, it is helpful to draw the figure.
In the two preceding examples, the number in the radical in the Quadratic Formula was a perfect square and so the solutions were rational numbers. If we get an irrational number as a solution to an application problem, we will use a calculator to get an approximate value.
We will use the formula for the area of a rectangle to solve the next example.
The Pythagorean Theorem gives the relation between the legs and hypotenuse of a right triangle. We will use the Pythagorean Theorem to solve the next example.
The height of a projectile shot upward from the ground is modeled by a quadratic equation. The initial velocity, v0, propels the object up until gravity causes the object to fall back down.
We can use this formula to find how many seconds it will take for a firework to reach a specific height.
We have solved uniform motion problems using the formula D = rt in previous chapters. We used a table like the one below to organize the information and lead us to the equation.
The formula D = rt 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
Some uniform motion problems are also modeled by quadratic equations.
Work applications can also be modeled by quadratic equations. We will set them up using the same methods we used when we solved them with rational equations.We’ll use a similar scenario now.
### Key Concepts
1. Methods to Solve Quadratic Equations
2. How to use a Problem-Solving Strategy.
3. Area of a Triangle
4. Area of a Rectangle
5. Pythagorean Theorem
6. Projectile motion
### Practice Makes Pefect
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve using any method.
In the following exercises, solve using any method. Round your answers to the nearest tenth, if needed.
### 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? |
# Quadratic Equations and Functions
## Graph Quadratic Functions Using Properties
### Recognize the Graph of a Quadratic Function
Previously we very briefly looked at the function , which we called the square function. It was one of the first non-linear functions we looked at. Now we will graph functions of the form if We call this kind of function a quadratic function.
We graphed the quadratic function by plotting points.
Every quadratic function has a graph that looks like this. We call this figure a parabola.
Let’s practice graphing a parabola by plotting a few points.
All graphs of quadratic functions of the form f (x) = ax2 + bx + c are parabolas that open upward or downward. See .
Notice that the only difference in the two functions is the negative sign before the quadratic term (x2 in the equation of the graph in ). When the quadratic term, is positive, the parabola opens upward, and when the quadratic term is negative, the parabola opens downward.
### Find the Axis of Symmetry and Vertex of a Parabola
Look again at . Do you see that we could fold each parabola in half and then one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.
We show the same two graphs again with the axis of symmetry. See .
The equation of the axis of symmetry can be derived by using the Quadratic Formula. We will omit the derivation here and proceed directly to using the result. The equation of the axis of symmetry of the graph of f (x) = ax2 + bx + c is
So to find the equation of symmetry of each of the parabolas we graphed above, we will substitute into the formula
Notice that these are the equations of the dashed blue lines on the graphs.
The point on the parabola that is the lowest (parabola opens up), or the highest (parabola opens down), lies on the axis of symmetry. This point is called the vertex of the parabola.
We can easily find the coordinates of the vertex, because we know it is on the axis of symmetry. This means itsx-coordinate is To find the y-coordinate of the vertex we substitute the value of the x-coordinate into the quadratic function.
### Find the Intercepts of a Parabola
When we graphed linear equations, we often used the x- and y-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.
Remember, at the y-intercept the value of x is zero. So to find the y-intercept, we substitute x = 0 into the function.
Let’s find the y-intercepts of the two parabolas shown in .
An x-intercept results when the value of f (x) is zero. To find an x-intercept, we let f (x) = 0. In other words, we will need to solve the equation 0 = ax2 + bx + c for x.
Solving quadratic equations like this is exactly what we have done earlier in this chapter!
We can now find the x-intercepts of the two parabolas we looked at. First we will find the x-intercepts of the parabola whose function is f (x) = x2 + 4x + 3.
Now we will find the x-intercepts of the parabola whose function is f (x) = −x2 + 4x + 3.
We will use the decimal approximations of the x-intercepts, so that we can locate these points on the graph,
Do these results agree with our graphs? See .
In this chapter, we have been solving quadratic equations of the form ax2 + bx + c = 0. We solved for x and the results were the solutions to the equation.
We are now looking at quadratic functions of the form f (x) = ax2 + bx + c. The graphs of these functions are parabolas. The x-intercepts of the parabolas occur where f (x) = 0.
For example:
The solutions of the quadratic function are the x values of the x-intercepts.
Earlier, we saw that quadratic equations have 2, 1, or 0 solutions. The graphs below show examples of parabolas for these three cases. Since the solutions of the functions give the x-intercepts of the graphs, the number of x-intercepts is the same as the number of solutions.
Previously, we used the discriminant to determine the number of solutions of a quadratic function of the form Now we can use the discriminant to tell us how many x-intercepts there are on the graph.
Before you to find the values of the x-intercepts, you may want to evaluate the discriminant so you know how many solutions to expect.
### Graph Quadratic Functions Using Properties
Now we have all the pieces we need in order to graph a quadratic function. We just need to put them together. In the next example we will see how to do this.
We list the steps to take in order to graph a quadratic function here.
We were able to find the x-intercepts in the last example by factoring. We find the x-intercepts in the next example by factoring, too.
For the graph of f (x) = −x2 + 6x − 9, the vertex and the x-intercept were the same point. Remember how the discriminant determines the number of solutions of a quadratic equation? The discriminant of the equation 0 = −x2 + 6x − 9 is 0, so there is only one solution. That means there is only one x-intercept, and it is the vertex of the parabola.
How many x-intercepts would you expect to see on the graph of f (x) = x2 + 4x + 5?
Finding the y-intercept by finding f (0) is easy, isn’t it? Sometimes we need to use the Quadratic Formula to find the x-intercepts.
### Solve Maximum and Minimum Applications
Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic function. The y-coordinate of the vertex is the minimum value of a parabola that opens upward. It is the maximum value of a parabola that opens downward. See .
We have used the formula
to calculate the height in feet, h , of an object shot upwards into the air with initial velocity, v0, after t seconds .
This formula is a quadratic function, so its graph is a parabola. By solving for the coordinates of the vertex (t, h), we can find how long it will take the object to reach its maximum height. Then we can calculate the maximum height.
### Key Concepts
1. Parabola Orientation
2. Axis of Symmetry and Vertex of a Parabola The graph of the function is a parabola where:
3. Find the Intercepts of a Parabola
4. How to graph a quadratic function using properties.
5. Minimum or Maximum Values of a Quadratic Equation
### Practice Makes Perfect
Recognize the Graph of a Quadratic Function
In the following exercises, graph the functions by plotting points.
For each of the following exercises, determine if the parabola opens up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.
Find the Intercepts of a Parabola
In the following exercises, find the intercepts of the parabola whose function is given.
Graph Quadratic Functions Using Properties
In the following exercises, graph the function by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the maximum or minimum value of each function.
In the following exercises, solve. Round answers to the nearest tenth.
### Writing Exercise
### 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? |
# Quadratic Equations and Functions
## Graph Quadratic Functions Using Transformations
### Graph Quadratic Functions of the form
In the last section, we learned how to graph quadratic functions using their properties. Another method involves starting with the basic graph of and ‘moving’ it according to information given in the function equation. We call this graphing quadratic functions using transformations.
In the first example, we will graph the quadratic function by plotting points. Then we will see what effect adding a constant, k, to the equation will have on the graph of the new function
The last example shows us that to graph a quadratic function of the form we take the basic parabola graph of and vertically shift it up or shift it down .
This transformation is called a vertical shift.
Now that we have seen the effect of the constant, k, it is easy to graph functions of the form We just start with the basic parabola of and then shift it up or down.
It may be helpful to practice sketching quickly. We know the values and can sketch the graph from there.
Once we know this parabola, it will be easy to apply the transformations. The next example will require a vertical shift.
### Graph Quadratic Functions of the form
In the first example, we graphed the quadratic function by plotting points and then saw the effect of adding a constant k to the function had on the resulting graph of the new function
We will now explore the effect of subtracting a constant, h, from x has on the resulting graph of the new function
The last example shows us that to graph a quadratic function of the form we take the basic parabola graph of and shift it left (h > 0) or shift it right (h < 0).
This transformation is called a horizontal shift.
Now that we have seen the effect of the constant, h, it is easy to graph functions of the form We just start with the basic parabola of and then shift it left or right.
The next example will require a horizontal shift.
Now that we know the effect of the constants h and k, we will graph a quadratic function of the form by first drawing the basic parabola and then making a horizontal shift followed by a vertical shift. We could do the vertical shift followed by the horizontal shift, but most students prefer the horizontal shift followed by the vertical.
### Graph Quadratic Functions of the Form
So far we graphed the quadratic function and then saw the effect of including a constant h or k in the equation had on the resulting graph of the new function. We will now explore the effect of the coefficient a on the resulting graph of the new function
If we graph these functions, we can see the effect of the constant a, assuming a > 0.
To graph a function with constant a it is easiest to choose a few points on and multiply the y-values by a.
### Graph Quadratic Functions Using Transformations
We have learned how the constants a, h, and k in the functions, and affect their graphs. We can now put this together and graph quadratic functions by first putting them into the form by completing the square. This form is sometimes known as the vertex form or standard form.
We must be careful to both add and subtract the number to the SAME side of the function to complete the square. We cannot add the number to both sides as we did when we completed the square with quadratic equations.
When we complete the square in a function with a coefficient of x2 that is not one, we have to factor that coefficient from just the x-terms. We do not factor it from the constant term. It is often helpful to move the constant term a bit to the right to make it easier to focus only on the x-terms.
Once we get the constant we want to complete the square, we must remember to multiply it by that coefficient before we then subtract it.
Once we put the function into the form, we can then use the transformations as we did in the last few problems. The next example will show us how to do this.
We list the steps to take to graph a quadratic function using transformations here.
Now that we have completed the square to put a quadratic function into form, we can also use this technique to graph the function using its properties as in the previous section.
If we look back at the last few examples, we see that the vertex is related to the constants h and k.
In each case, the vertex is (h, k). Also the axis of symmetry is the line x = h.
We rewrite our steps for graphing a quadratic function using properties for when the function is in form.
### Find a Quadratic Function from its Graph
So far we have started with a function and then found its graph.
Now we are going to reverse the process. Starting with the graph, we will find the function.
### Key Concepts
1. Graph a Quadratic Function of the form Using a Vertical Shift
2. Graph a Quadratic Function of the form Using a Horizontal Shift
3. Graph of a Quadratic Function of the form
4. How to graph a quadratic function using transformations
5. Graph a quadratic function in the vertex form using properties
### Practice Makes Perfect
Graph Quadratic Functions of the form
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, k, to the function has on the basic parabola.
In the following exercises, graph each function using a vertical shift.
Graph Quadratic Functions of the form
In the following exercises, ⓐ graph the quadratic functions on the same rectangular coordinate system and ⓑ describe what effect adding a constant, , inside the parentheses has
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the form by completing the square.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it using properties.
Matching
In the following exercises, match the graphs to one of the following functions: ⓐ ⓑ ⓒ ⓓ ⓔ ⓕ ⓖ ⓗ
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in form whose graph is shown.
### Writing Exercise
### 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? |
# Quadratic Equations and Functions
## Solve Quadratic Inequalities
We have learned how to solve linear inequalities and rational inequalities previously. Some of the techniques we used to solve them were the same and some were different.
We will now learn to solve inequalities that have a quadratic expression. We will use some of the techniques from solving linear and rational inequalities as well as quadratic equations.
We will solve quadratic inequalities two ways—both graphically and algebraically.
### Solve Quadratic Inequalities Graphically
A quadratic equation is in standard form when written as ax2 + bx + c = 0. If we replace the equal sign with an inequality sign, we have a quadratic inequality in standard form.
The graph of a quadratic function f(x) = ax2 + bx + c = 0 is a parabola. When we ask when is ax2 + bx + c < 0, we are asking when is f(x) < 0. We want to know when the parabola is below the x-axis.
When we ask when is ax2 + bx + c > 0, we are asking when is f(x) > 0. We want to know when the parabola is above the x-axis.
We list the steps to take to solve a quadratic inequality graphically.
In the last example, the parabola opened upward and in the next example, it opens downward. In both cases, we are looking for the part of the parabola that is below the x-axis but note how the position of the parabola affects the solution.
### Solve Quadratic Inequalities Algebraically
The algebraic method we will use is very similar to the method we used to solve rational inequalities. We will find the critical points for the inequality, which will be the solutions to the related quadratic equation. Remember a polynomial expression can change signs only where the expression is zero.
We will use the critical points to divide the number line into intervals and then determine whether the quadratic expression willl be postive or negative in the interval. We then determine the solution for the inequality.
In this example, since the expression factors nicely, we can also find the sign in each interval much like we did when we solved rational inequalities. We find the sign of each of the factors, and then the sign of the product. Our number line would like this:
The result is the same as we found using the other method.
We summarize the steps here.
The solutions of the quadratic inequalities in each of the previous examples, were either an interval or the union of two intervals. This resulted from the fact that, in each case we found two solutions to the corresponding quadratic equation ax2 + bx + c = 0. These two solutions then gave us either the two x-intercepts for the graph or the two critical points to divide the number line into intervals.
This correlates to our previous discussion of the number and type of solutions to a quadratic equation using the discriminant.
For a quadratic equation of the form ax2 + bx + c = 0,
The last row of the table shows us when the parabolas never intersect the x-axis. Using the Quadratic Formula to solve the quadratic equation, the radicand is a negative. We get two complex solutions.
In the next example, the quadratic inequality solutions will result from the solution of the quadratic equation being complex.
### Key Concepts
1. Solve a Quadratic Inequality Graphically
2. How to Solve a Quadratic Inequality Algebraically
### Section Exercises
### Practice Makes Perfect
Solve Quadratic Inequalities Graphically
In the following exercises, ⓐ solve graphically and ⓑ write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
### 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?
### Chapter Review Exercises
### Solve Quadratic Equations Using the Square Root Property
Solve Quadratic Equations of the form
In the following exercises, solve using the Square Root Property.
Solve Quadratic Equations of the Form
In the following exercises, solve using the Square Root Property.
### Solve Quadratic Equations by Completing the Square
Solve Quadratic Equations Using Completing the Square
In the following exercises, complete the square to make a perfect square trinomial. Then write the result as a binomial squared.
In the following exercises, solve by completing the square.
Solve Quadratic Equations of the form
In the following exercises, solve by completing the square.
### 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 for 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.
### Solve Equations in Quadratic Form
Solve Equations in Quadratic Form
In the following exercises, solve.
### Solve Applications of Quadratic Equations
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve by using the method of factoring, the square root principle, or the Quadratic Formula. Round your answers to the nearest tenth, if needed.
### Graph Quadratic Functions Using Properties
Recognize the Graph of a Quadratic Function
In the following exercises, graph by plotting point.
In the following exercises, determine if the following parabolas open up or down.
Find the Axis of Symmetry and Vertex of a Parabola
In the following exercises, find ⓐ the equation of the axis of symmetry and ⓑ the vertex.
Find the Intercepts of a Parabola
In the following exercises, find the x- and y-intercepts.
Graph Quadratic Functions Using Properties
In the following exercises, graph by using its properties.
Solve Maximum and Minimum Applications
In the following exercises, find the minimum or maximum value.
In the following exercises, solve. Rounding answers to the nearest tenth.
### Graph Quadratic Functions Using Transformations
Graph Quadratic Functions of the form
In the following exercises, graph each function using a vertical shift.
In the following exercises, graph each function using a horizontal shift.
In the following exercises, graph each function using transformations.
Graph Quadratic Functions of the form
In the following exercises, graph each function.
Graph Quadratic Functions Using Transformations
In the following exercises, rewrite each function in the form by completing the square.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it by using transformations.
In the following exercises, ⓐ rewrite each function in form and ⓑ graph it using properties.
Find a Quadratic Function from its Graph
In the following exercises, write the quadratic function in form.
### Solve Quadratic Inequalities
Solve Quadratic Inequalities Graphically
In the following exercises, solve graphically and write the solution in interval notation.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
### Practice Test
Solve the following quadratic equations. Use any method.
Use the discriminant to determine the number and type of solutions of each quadratic equation.
Solve each equation.
For each parabola, find ⓐ which direction it opens, ⓑ the equation of the axis of symmetry, ⓒ the vertex, ⓓ the x- and y-intercepts, and e) the maximum or minimum value.
Graph each quadratic function using intercepts, the vertex, and the equation of the axis of symmetry.
In the following exercises, graph each function using transformations.
In the following exercises, solve each inequality algebraically and write any solution in interval notation.
Model the situation with a quadratic equation and solve by any method. |
# Exponential and Logarithmic Functions
## Introduction
As the world population continues to grow, food supplies are becoming less able to meet the increasing demand. At the same time, available resources of fertile soil for growing plants is dwindling. One possible solution—grow plants without soil. Botanists around the world are expanding the potential of hydroponics, which is the process of growing plants without soil. To provide the plants with the nutrients they need, the botanists keep careful growth records. Some growth is described by the types of functions you will explore in this chapter—exponential and logarithmic. You will evaluate and graph these functions, and solve equations using them. |
# Exponential and Logarithmic Functions
## Finding Composite and Inverse Functions
In this chapter, we will introduce two new types of functions, exponential functions and logarithmic functions. These functions are used extensively in business and the sciences as we will see.
### Find and Evaluate Composite Functions
Before we introduce the functions, we need to look at another operation on functions called composition. In composition, the output of one function is the input of a second function. For functions and the composition is written and is defined by
We read as of of
To do a composition, the output of the first function, becomes the input of the second function, f, and so we must be sure that it is part of the domain of f.
We have actually used composition without using the notation many times before. When we graphed quadratic functions using translations, we were composing functions. For example, if we first graphed as a parabola and then shifted it down vertically four units, we were using the composition defined by where
The next example will demonstrate that and usually result in different outputs.
In the next example we will evaluate a composition for a specific value.
### Determine Whether a Function is One-to-One
When we first introduced functions, we said a function is a relation that assigns to each element in its domain exactly one element in the range. For each ordered pair in the relation, each x-value is matched with only one y-value.
We used the birthday example to help us understand the definition. Every person has a birthday, but no one has two birthdays and it is okay for two people to share a birthday. Since each person has exactly one birthday, that relation is a function.
A function is one-to-one if each value in the range has exactly one element in the domain. For each ordered pair in the function, each y-value is matched with only one x-value.
Our example of the birthday relation is not a one-to-one function. Two people can share the same birthday. The range value August 2 is the birthday of Liz and June, and so one range value has two domain values. Therefore, the function is not one-to-one.
To help us determine whether a relation is a function, we use the vertical line test. A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point. Also, if any vertical line intersects the graph in more than one point, the graph does not represent a function.
The vertical line is representing an x-value and we check that it intersects the graph in only one y-value. Then it is a function.
To check if a function is one-to-one, we use a similar process. We use a horizontal line and check that each horizontal line intersects the graph in only one point. The horizontal line is representing a y-value and we check that it intersects the graph in only one x-value. If every horizontal line intersects the graph of a function in at most one point, it is a one-to-one function. This is the horizontal line test.
We can test whether a graph of a relation is a function by using the vertical line test. We can then tell if the function is one-to-one by applying the horizontal line test.
### Find the Inverse of a Function
Let’s look at a one-to one function, , represented by the ordered pairs For each -value, adds 5 to get the -value. To ‘undo’ the addition of 5, we subtract 5 from each -value and get back to the original -value. We can call this “taking the inverse of ” and name the function
Notice that that the ordered pairs of and have their -values and -values reversed. The domain of is the range of and the domain of is the range of
In the next example we will find the inverse of a function defined by ordered pairs.
We just noted that if is a one-to-one function whose ordered pairs are of the form then its inverse function is the set of ordered pairs
So if a point is on the graph of a function then the ordered pair is on the graph of See .
The distance between any two pairs and is cut in half by the line So we say the points are mirror images of each other through the line
Since every point on the graph of a function is a mirror image of a point on the graph of we say the graphs are mirror images of each other through the line We will use this concept to graph the inverse of a function in the next example.
When we began our discussion of an inverse function, we talked about how the inverse function ‘undoes’ what the original function did to a value in its domain in order to get back to the original x-value.
We can use this property to verify that two functions are inverses of each other.
We have found inverses of function defined by ordered pairs and from a graph. We will now look at how to find an inverse using an algebraic equation. The method uses the idea that if is a one-to-one function with ordered pairs then its inverse function is the set of ordered pairs
If we reverse the x and y in the function and then solve for y, we get our inverse function.
We summarize the steps below.
### Key Concepts
1. Composition of Functions: The composition of functions and is written and is defined by
We read as of of
2. Horizontal Line Test: If every horizontal line, intersects the graph of a function in at most one point, it is a one-to-one function.
3. Inverse of a Function Defined by Ordered Pairs: If is a one-to-one function whose ordered pairs are of the form then its inverse function is the set of ordered pairs
4. Inverse Functions: For every in the domain of one-to-one function and
5. How to Find the Inverse of a One-to-One Function:
### Practice Makes Perfect
Find and Evaluate Composite Functions
In the following exercises, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).
In the following exercises, find the values described.
Determine Whether a Function is One-to-One
In the following exercises, determine if the set of ordered pairs represents a function and if so, is the function one-to-one.
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
In the following exercises, find the inverse of each function. Determine the domain and range of the inverse function.
In the following exercises, graph, on the same coordinate system, the inverse of the one-to-one function shown.
In the following exercises, determine whether or not the given functions are inverses.
In the following exercises, find the inverse of each function.
### 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. |
# Exponential and Logarithmic Functions
## Evaluate and Graph Exponential Functions
### Graph Exponential Functions
The functions we have studied so far do not give us a model for many naturally occurring phenomena. From the growth of populations and the spread of viruses to radioactive decay and compounding interest, the models are very different from what we have studied so far. These models involve exponential functions.
An exponential function is a function of the form where and
Notice that in this function, the variable is the exponent. In our functions so far, the variables were the base.
Our definition says If we let then becomes Since for all real numbers, This is the constant function.
Our definition also says If we let a base be negative, say then is not a real number when
In fact, would not be a real number any time is a fraction with an even denominator. So our definition requires
By graphing a few exponential functions, we will be able to see their unique properties.
If we look at the graphs from the previous Example and Try Its, we can identify some of the properties of exponential functions.
The graphs of and as well as the graphs of and all have the same basic shape. This is the shape we expect from an exponential function where
We notice, that for each function, the graph contains the point This make sense because for any a.
The graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as
Notice too, the graph of each function also contains the point The graph of contained and the graph of contained This makes sense as
What is the domain for each function? From the graphs we can see that the domain is the set of all real numbers. There is no restriction on the domain. We write the domain in interval notation as
Look at each graph. What is the range of the function? The graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as
Whenever a graph of a function approaches a line but never touches it, we call that line an asymptote. For the exponential functions we are looking at, the graph approaches the -axis very closely but will never cross it, we call the line the x-axis, a horizontal asymptote.
Our definition of an exponential function says but the examples and discussion so far has been about functions where What happens when ? The next example will explore this possibility.
Now let’s look at the graphs from the previous Example and Try Its so we can now identify some of the properties of exponential functions where
The graphs of and as well as the graphs of and all have the same basic shape. While this is the shape we expect from an exponential function where the graphs go down from left to right while the previous graphs, when went from up from left to right.
We notice that for each function, the graph still contains the point (0, 1). This make sense because for any a.
As before, the graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as
Notice too that the graph of each function, also contains the point The graph of contained and the graph of contained This makes sense as
What is the domain and range for each function? From the graphs we can see that the domain is the set of all real numbers and we write the domain in interval notation as Again, the graph never hits the -axis. The range is all positive numbers. We write the range in interval notation as
We will summarize these properties in the chart below. Which also include when
It is important for us to notice that both of these graphs are one-to-one, as they both pass the horizontal line test. This means the exponential function will have an inverse. We will look at this later.
When we graphed quadratic functions, we were able to graph using translation rather than just plotting points. Will that work in graphing exponential functions?
Looking at the graphs of the functions and in the last example, we see that adding one in the exponent caused a horizontal shift of one unit to the left. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
Let’s now consider another situation that might be graphed more easily by translation, once we recognize the pattern.
Looking at the graphs of the functions and in the last example, we see that subtracting 2 caused a vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. Recognizing this pattern allows us to graph other functions with the same pattern by translation.
All of our exponential functions have had either an integer or a rational number as the base. We will now look at an exponential function with an irrational number as the base.
Before we can look at this exponential function, we need to define the irrational number, e. This number is used as a base in many applications in the sciences and business that are modeled by exponential functions. The number is defined as the value of as n gets larger and larger. We say, as n approaches infinity, or increases without bound. The table shows the value of for several values of
The number e is like the number in that we use a symbol to represent it because its decimal representation never stops or repeats. The irrational number e is called the natural base.
The exponential function whose base is is called the natural exponential function.
Let’s graph the function on the same coordinate system as and
Notice that the graph of is “between” the graphs of and Does this make sense as ?
### Solve Exponential Equations
Equations that include an exponential expression are called exponential equations. To solve them we use a property that says as long as and if then it is true that In other words, in an exponential equation, if the bases are equal then the exponents are equal.
To use this property, we must be certain that both sides of the equation are written with the same base.
The steps are summarized below.
In the next example, we will use our properties on exponents.
### Use Exponential Models in Applications
Exponential functions model many situations. If you own a bank account, you have experienced the use of an exponential function. There are two formulas that are used to determine the balance in the account when interest is earned. If a principal, P, is invested at an interest rate, r, for t years, the new balance, A, will depend on how often the interest is compounded. If the interest is compounded n times a year we use the formula If the interest is compounded continuously, we use the formula These are the formulas for compound interest.
As you work with the Interest formulas, it is often helpful to identify the values of the variables first and then substitute them into the formula.
Other topics that are modeled by exponential functions involve growth and decay. Both also use the formula we used for the growth of money. For growth and decay, generally we use as the original amount instead of calling it the principal. We see that exponential growth has a positive rate of growth and exponential decay has a negative rate of growth.
Exponential growth is typically seen in the growth of populations of humans or animals or bacteria. Our next example looks at the growth of a virus.
### Key Concepts
1. Properties of the Graph of
2. One-to-One Property of Exponential Functions:
For and
3. How to Solve an Exponential Equation
4. Compound Interest: For a principal, invested at an interest rate, for years, the new balance, is
5. Exponential Growth and Decay: For an original amount, that grows or decays at a rate, for a certain time the final amount, is
### Practice Makes Perfect
Graph Exponential Functions
In the following exercises, graph each exponential function.
In the following exercises, graph each function in the same coordinate system.
In the following exercises, graph each exponential function.
Solve Exponential Equations
In the following exercises, solve each equation.
In the following exercises, match the graphs to one of the following functions: ⓐ ⓑ ⓒ ⓓ ⓔ ⓕ
Use exponential models in applications
In the following exercises, use an exponential model to solve.
### 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? |
# Exponential and Logarithmic Functions
## Evaluate and Graph Logarithmic Functions
We have spent some time finding the inverse of many functions. It works well to ‘undo’ an operation with another operation. Subtracting ‘undoes’ addition, multiplication ‘undoes’ division, taking the square root ‘undoes’ squaring.
As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.
To deal with this we define the logarithm function with base a to be the inverse of the exponential function We use the notation and say the inverse function of the exponential function is the logarithmic function.
### Convert Between Exponential and Logarithmic Form
Since the equations and are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See . Notice the positions of the exponent and base.
If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.”
In the next example we do the reverse—convert logarithmic form to exponential form.
### Evaluate Logarithmic Functions
We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.
When see an expression such as we can find its exact value two ways. By inspection we realize it means to what power will be Since we know An alternate way is to set the expression equal to and then convert it into an exponential equation.
### Graph Logarithmic Functions
To graph a logarithmic function it is easiest to convert the equation to its exponential form, Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.
The graphs of and are the shape we expect from a logarithmic function where
We notice that for each function the graph contains the point This make sense because means which is true for any a.
The graph of each function, also contains the point This makes sense as means which is true for any a.
Notice too, the graph of each function also contains the point This makes sense as means which is true for any a.
Look at each graph again. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function.
What is the domain of the function? The graph never hits the y-axis. The domain is all positive numbers. We write the domain in interval notation as
What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as
When the graph approaches the y-axis so very closely but will never cross it, we call the line the y-axis, a vertical asymptote.
Our next example looks at the graph of when
Now, let’s look at the graphs and , so we can identify some of the properties of logarithmic functions where
The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where
We notice, that for each function again, the graph contains the points, This make sense for the same reasons we argued above.
We notice the domain and range are also the same—the domain is and the range is The -axis is again the vertical asymptote.
We will summarize these properties in the chart below. Which also include when
We talked earlier about how the logarithmic function is the inverse of the exponential function The graphs in show both the exponential (blue) and logarithmic (red) functions on the same graph for both and
Notice how the graphs are reflections of each other through the line We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the x-axis is the horizontal asymptote for the exponential functions and the y-axis is the vertical asymptote for the logarithmic functions.
### Solve Logarithmic Equations
When we talked about exponential functions, we introduced the number e. Just as e was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base e is called the natural logarithmic function. The function is generally written and we read it as “el en of
When the base of the logarithm function is 10, we call it the common logarithmic function and the base is not shown. If the base a of a logarithm is not shown, we assume it is 10.
To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, , we need to remember that for the base a, and Also, the domain is Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.
### Use Logarithmic Models in Applications
There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from 0, which is barely audible to 160, which can rupture an eardrum. The in the formula represents the intensity of sound that is barely audible.
The magnitude of an earthquake is measured by a logarithmic scale called the Richter scale. The model is where is the intensity of the shock wave. This model provides a way to measure earthquake intensity.
### Key Concepts
1. Properties of the Graph of
2. Decibel Level of Sound: The loudness level, , measured in decibels, of a sound of intensity, , measured in watts per square inch is
3. Earthquake Intensity: The magnitude of an earthquake is measured by where is the intensity of its shock wave.
### Practice Makes Perfect
Convert Between Exponential and Logarithmic Form
In the following exercises, convert from exponential to logarithmic form.
In the following exercises, convert each logarithmic equation to exponential form.
Evaluate Logarithmic Functions
In the following exercises, find the value of in each logarithmic equation.
In the following exercises, find the exact value of each logarithm without using a calculator.
Graph Logarithmic Functions
In the following exercises, graph each logarithmic function.
Solve Logarithmic Equations
In the following exercises, solve each logarithmic equation.
Use Logarithmic Models in Applications
In the following exercises, use a logarithmic model to solve.
### 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? |
# Exponential and Logarithmic Functions
## Use the Properties of Logarithms
### Use the Properties of Logarithms
Now that we have learned about exponential and logarithmic functions, we can introduce some of the properties of logarithms. These will be very helpful as we continue to solve both exponential and logarithmic equations.
The first two properties derive from the definition of logarithms. Since we can convert this to logarithmic form and get Also, since we get
In the next example we could evaluate the logarithm by converting to exponential form, as we have done previously, but recognizing and then applying the properties saves time.
The next two properties can also be verified by converting them from exponential form to logarithmic form, or the reverse.
The exponential equation converts to the logarithmic equation which is a true statement for positive values for x only.
The logarithmic equation converts to the exponential equation which is also a true statement.
These two properties are called inverse properties because, when we have the same base, raising to a power “undoes” the log and taking the log “undoes” raising to a power. These two properties show the composition of functions. Both ended up with the identity function which shows again that the exponential and logarithmic functions are inverse functions.
In the next example, apply the inverse properties of logarithms.
There are three more properties of logarithms that will be useful in our work. We know exponential functions and logarithmic function are very interrelated. Our definition of logarithm shows us that a logarithm is the exponent of the equivalent exponential. The properties of exponents have related properties for exponents.
In the Product Property of Exponents, we see that to multiply the same base, we add the exponents. The Product Property of Logarithms, tells us to take the log of a product, we add the log of the factors.
We use this property to write the log of a product as a sum of the logs of each factor.
Similarly, in the Quotient Property of Exponents, we see that to divide the same base, we subtract the exponents. The Quotient Property of Logarithms, tells us to take the log of a quotient, we subtract the log of the numerator and denominator.
Note that
We use this property to write the log of a quotient as a difference of the logs of each factor.
The third property of logarithms is related to the Power Property of Exponents, we see that to raise a power to a power, we multiply the exponents. The Power Property of Logarithms, tells us to take the log of a number raised to a power, we multiply the power times the log of the number.
We use this property to write the log of a number raised to a power as the product of the power times the log of the number. We essentially take the exponent and throw it in front of the logarithm.
We summarize the Properties of Logarithms here for easy reference. While the natural logarithms are a special case of these properties, it is often helpful to also show the natural logarithm version of each property.
Now that we have the properties we can use them to “expand” a logarithmic expression. This means to write the logarithm as a sum or difference and without any powers.
We generally apply the Product and Quotient Properties before we apply the Power Property.
When we have a radical in the logarithmic expression, it is helpful to first write its radicand as a rational exponent.
The opposite of expanding a logarithm is to condense a sum or difference of logarithms that have the same base into a single logarithm. We again use the properties of logarithms to help us, but in reverse.
To condense logarithmic expressions with the same base into one logarithm, we start by using the Power Property to get the coefficients of the log terms to be one and then the Product and Quotient Properties as needed.
### Use the Change-of-Base Formula
To evaluate a logarithm with any other base, we can use the Change-of-Base Formula. We will show how this is derived.
The Change-of-Base Formula introduces a new base This can be any base b we want where Because our calculators have keys for logarithms base 10 and base e, we will rewrite the Change-of-Base Formula with the new base as 10 or e.
When we use a calculator to find the logarithm value, we usually round to three decimal places. This gives us an approximate value and so we use the approximately equal symbol .
### Key Concepts
1. Properties of Logarithms
2. Inverse Properties of Logarithms
3. Product Property of Logarithms
4. Quotient Property of Logarithms
5. Power Property of Logarithms
6. Properties of Logarithms Summary
If and is any real number then,
7. Change-of-Base Formula
For any logarithmic bases a and b, and
### Practice Makes Perfect
Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Power Property of Logarithms to expand each. Simplify if possible.
In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
Use the Change-of-Base Formula
In the following exercises, use the Change-of-Base Formula, rounding to three decimal places, to approximate each logarithm.
### 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 how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this? |
# Exponential and Logarithmic Functions
## Solve Exponential and Logarithmic Equations
### Solve Logarithmic Equations Using the Properties of Logarithms
In the section on logarithmic functions, we solved some equations by rewriting the equation in exponential form. Now that we have the properties of logarithms, we have additional methods we can use to solve logarithmic equations.
If our equation has two logarithms we can use a property that says that if then it is true that This is the One-to-One Property of Logarithmic Equations.
To use this property, we must be certain that both sides of the equation are written with the same base.
Remember that logarithms are defined only for positive real numbers. Check your results in the original equation. You may have obtained a result that gives a logarithm of zero or a negative number.
Another strategy to use to solve logarithmic equations is to condense sums or differences into a single logarithm.
When there are logarithms on both sides, we condense each side into a single logarithm. Remember to use the Power Property as needed.
### Solve Exponential Equations Using Logarithms
In the section on exponential functions, we solved some equations by writing both sides of the equation with the same base. Next we wrote a new equation by setting the exponents equal.
It is not always possible or convenient to write the expressions with the same base. In that case we often take the common logarithm or natural logarithm of both sides once the exponential is isolated.
When we take the logarithm of both sides we will get the same result whether we use the common or the natural logarithm (try using the natural log in the last example. Did you get the same result?) When the exponential has base e, we use the natural logarithm.
### Use Exponential Models in Applications
In previous sections we were able to solve some applications that were modeled with exponential equations. Now that we have so many more options to solve these equations, we are able to solve more applications.
We will again use the Compound Interest Formulas and so we list them here for reference.
We have seen that growth and decay are modeled by exponential functions. For growth and decay we use the formula Exponential growth has a positive rate of growth or growth constant, , and exponential decay has a negative rate of growth or decay constant, k.
We can now solve applications that give us enough information to determine the rate of growth. We can then use that rate of growth to predict other situations.
Radioactive substances decay or decompose according to the exponential decay formula. The amount of time it takes for the substance to decay to half of its original amount is called the half-life of the substance.
Similar to the previous example, we can use the given information to determine the constant of decay, and then use that constant to answer other questions.
### Key Concepts
1. One-to-One Property of Logarithmic Equations: For and is any real number:
2. Compound Interest:
For a principal, P, invested at an interest rate, r, for t years, the new balance, A, is:
3. Exponential Growth and Decay: For an original amount, that grows or decays at a rate, r, for a certain time t, the final amount, A, is
### Section Exercises
### Practice Makes Perfect
Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
In the following exercises, solve each equation.
In the following exercises, solve for x, giving an exact answer as well as an approximation to three decimal places.
Use Exponential Models in Applications
In the following exercises, 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?
### Chapter Review Exercises
### Finding Composite and Inverse Functions
Find and Evaluate Composite Functions
In the following exercises, for each pair of functions, find ⓐ (f ∘ g)(x), ⓑ (g ∘ f)(x), and ⓒ (f · g)(x).
In the following exercises, evaluate the composition.
Determine Whether a Function is One-to-One
In the following exercises, for each set of ordered pairs, determine if it represents a function and if so, is the function one-to-one.
In the following exercises, determine whether each graph is the graph of a function and if so, is it one-to-one.
Find the Inverse of a Function
In the following exercise, find the inverse of the function. Determine the domain and range of the inverse function.
In the following exercise, graph the inverse of the one-to-one function shown.
In the following exercises, verify that the functions are inverse functions.
In the following exercises, find the inverse of each function.
### Evaluate and Graph Exponential Functions
Graph Exponential Functions
In the following exercises, graph each of the following functions.
Solve Exponential Equations
In the following exercises, solve each equation.
Use Exponential Models in Applications
In the following exercises, solve.
### Evaluate and Graph Logarithmic Functions
Convert Between Exponential and Logarithmic Form
In the following exercises, convert from exponential to logarithmic form.
In the following exercises, convert each logarithmic equation to exponential form.
Evaluate Logarithmic Functions
In the following exercises, solve for x.
In the following exercises, find the exact value of each logarithm without using a calculator.
Graph Logarithmic Functions
In the following exercises, graph each logarithmic function.
Solve Logarithmic Equations
In the following exercises, solve each logarithmic equation.
Use Logarithmic Models in Applications
### Use the Properties of Logarithms
Use the Properties of Logarithms
In the following exercises, use the properties of logarithms to evaluate.
In the following exercises, use the Product Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify, if possible.
In the following exercises, use the Power Property of Logarithms to expand each logarithm. Simplify, if possible.
In the following exercises, use properties of logarithms to write each logarithm as a sum of logarithms. Simplify if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm. Simplify if possible.
Use the Change-of-Base Formula
In the following exercises, rounding to three decimal places, approximate each logarithm.
### Solve Exponential and Logarithmic Equations
Solve Logarithmic Equations Using the Properties of Logarithms
In the following exercises, solve for x.
Solve Exponential Equations Using Logarithms
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places.
Use Exponential Models in Applications
### Practice Test
In the following exercises, use properties of logarithms to write each expression as a sum of logarithms, simplifying if possible.
In the following exercises, use the Properties of Logarithms to condense the logarithm, simplifying if possible.
In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. |
# Conics
## Introduction
Five, Four. Three. Two. One. Lift off. The rocket launches off the ground headed toward space. Unmanned spaceships, and spaceships in general, are designed by aerospace engineers. These engineers are investigating reusable rockets that return safely to Earth to be used again. Someday, rockets may carry passengers to the International Space Station and beyond. One essential math concept for aerospace engineers is that of conics. In this chapter, you will learn about conics, including circles, parabolas, ellipses, and hyperbolas. Then you will use what you learn to investigate systems of nonlinear equations. |
# Conics
## Distance and Midpoint Formulas; Circles
In this chapter we will be looking at the conic sections, usually called the conics, and their properties. The conics are curves that result from a plane intersecting a double cone—two cones placed point-to-point. Each half of a double cone is called a nappe.
There are four conics—the circle, parabola, ellipse, and hyperbola. The next figure shows how the plane intersecting the double cone results in each curve.
Each of the curves has many applications that affect your daily life, from your cell phone to acoustics and navigation systems. In this section we will look at the properties of a circle.
### Use the Distance Formula
We have used the Pythagorean Theorem to find the lengths of the sides of a right triangle. Here we will use this theorem again to find distances on the rectangular coordinate system. By finding distance on the rectangular coordinate system, we can make a connection between the geometry of a conic and algebra—which opens up a world of opportunities for application.
Our first step is to develop a formula to find distances between points on the rectangular coordinate system. We will plot the points and create a right triangle much as we did when we found slope in Graphs and Functions. We then take it one step further and use the Pythagorean Theorem to find the length of the hypotenuse of the triangle—which is the distance between the points.
The method we used in the last example leads us to the formula to find the distance between the two points and
When we found the length of the horizontal leg we subtracted which is
When we found the length of the vertical leg we subtracted which is
If the triangle had been in a different position, we may have subtracted or The expressions and vary only in the sign of the resulting number. To get the positive value-since distance is positive- we can use absolute value. So to generalize we will say and
In the Pythagorean Theorem, we substitute the general expressions and rather than the numbers.
This is the Distance Formula we use to find the distance d between the two points and
### Use the Midpoint Formula
It is often useful to be able to find the midpoint of a segment. For example, if you have the endpoints of the diameter of a circle, you may want to find the center of the circle which is the midpoint of the diameter. To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.
Both the Distance Formula and the Midpoint Formula depend on two points, and It is easy to confuse which formula requires addition and which subtraction of the coordinates. If we remember where the formulas come from, it may be easier to remember the formulas.
### Write the Equation of a Circle in Standard Form
As we mentioned, our goal is to connect the geometry of a conic with algebra. By using the coordinate plane, we are able to do this easily.
We define a circle as all points in a plane that are a fixed distance from a given point in the plane. The given point is called the center, and the fixed distance is called the radius, r, of the circle.
This is the standard form of the equation of a circle with center, and radius, r.
In the last example, the center was Notice what happened to the equation. Whenever the center is the standard form becomes
In the next example, the radius is not given. To calculate the radius, we use the Distance Formula with the two given points.
### Graph a Circle
Any equation of the form is the standard form of the equation of a circle with center, and radius, r. We can then graph the circle on a rectangular coordinate system.
Note that the standard form calls for subtraction from x and y. In the next example, the equation has so we need to rewrite the addition as subtraction of a negative.
To find the center and radius, we must write the equation in standard form. In the next example, we must first get the coefficient of to be one.
If we expand the equation from , the equation of the circle looks very different.
This form of the equation is called the general form of the equation of the circle.
If we are given an equation in general form, we can change it to standard form by completing the squares in both x and y. Then we can graph the circle using its center and radius.
In the next example, there is a y-term and a -term. But notice that there is no x-term, only an -term. We have seen this before and know that it means h is 0. We will need to complete the square for the y terms, but not for the x terms.
### Key Concepts
1. Distance Formula: The distance d between the two points and is
2. Midpoint Formula: The midpoint of the line segment whose endpoints are the two points and is
To find the midpoint of a line segment, we find the average of the x-coordinates and the average of the y-coordinates of the endpoints.
3. Circle: A circle is all points in a plane that are a fixed distance from a fixed point in the plane. The given point is called the center, and the fixed distance is called the radius, r, of the circle.
4. Standard Form of the Equation a Circle: The standard form of the equation of a circle with center, and radius, r, is
5. General Form of the Equation of a Circle: The general form of the equation of a circle is
### Practice Makes Perfect
Use the Distance Formula
In the following exercises, find the distance between the points. Write the answer in exact form and then find the decimal approximation, rounded to the nearest tenth if needed.
Use the Midpoint Formula
In the following exercises, ⓐ find the midpoint of the line segments whose endpoints are given and ⓑ plot the endpoints and the midpoint on a rectangular coordinate system.
Write the Equation of a Circle in Standard Form
In the following exercises, write the standard form of the equation of the circle with the given radius and center
In the following exercises, write the standard form of the equation of the circle with the given radius and center
For the following exercises, write the standard form of the equation of the circle with the given center with point on the circle.
Graph a Circle
In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.
In the following exercises, ⓐ identify the center and radius and ⓑ graph.
### 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. |
# Conics
## Parabolas
### Graph Vertical Parabolas
The next conic section we will look at is a parabola. We define a parabola as all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
Previously, we learned to graph vertical parabolas from the general form or the standard form using properties. Those methods will also work here. We will summarize the properties here.
The graphs show what the parabolas look like when they open up or down. Their position in relation to the x- or y-axis is merely an example.
To graph a parabola from these forms, we used the following steps.
The next example reviews the method of graphing a parabola from the general form of its equation.
The next example reviews the method of graphing a parabola from the standard form of its equation,
### Graph Horizontal Parabolas
Our work so far has only dealt with parabolas that open up or down. We are now going to look at horizontal parabolas. These parabolas open either to the left or to the right. If we interchange the x and y in our previous equations for parabolas, we get the equations for the parabolas that open to the left or to the right.
The graphs show what the parabolas look like when they to the left or to the right. Their position in relation to the x- or y-axis is merely an example.
Looking at these parabolas, do their graphs represent a function? Since both graphs would fail the vertical line test, they do not represent a function.
To graph a parabola that opens to the left or to the right is basically the same as what we did for parabolas that open up or down, with the reversal of the x and y variables.
In the next example, the vertex is not the origin.
In , we see the relationship between the equation in standard form and the properties of the parabola. The How To box lists the steps for graphing a parabola in the standard form We will use this procedure in the next example.
In the next example, we notice the a is negative and so the parabola opens to the left.
The next example requires that we first put the equation in standard form and then use the properties.
### Solve Applications with Parabolas
Many architectural designs incorporate parabolas. It is not uncommon for bridges to be constructed using parabolas as we will see in the next example.
### Key Concepts
1. Parabola: A parabola is all points in a plane that are the same distance from a fixed point and a fixed line. The fixed point is called the focus, and the fixed line is called the directrix of the parabola.
2. How to graph vertical parabolas
3. How to graph horizontal parabolas
### Practice Makes Perfect
Graph Vertical Parabolas
In the following exercises, graph each equation by using properties.
In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.
Graph Horizontal Parabolas
In the following exercises, graph each equation by using properties.
In the following exercises, ⓐ write the equation in standard form and ⓑ use properties of the standard form to graph the equation.
Mixed Practice
In the following exercises, match each graph to one of the following equations: ⓐ x2 + y2 = 64 ⓑ x2 + y2 = 49ⓒ (x + 5)2 + (y + 2)2 = 4 ⓓ (x − 2)2 + (y − 3)2 = 9 ⓔ y = −x2 + 8x − 15 ⓕ y = 6x2 + 2x − 1
Solve Applications with Parabolas
### 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? |
# Conics
## Ellipses
### Graph an Ellipse with Center at the Origin
The next conic section we will look at is an ellipse. We define an ellipse as all points in a plane where the sum of the distances from two fixed points is constant. Each of the given points is called a focus of the ellipse.
We can draw an ellipse by taking some fixed length of flexible string and attaching the ends to two thumbtacks. We use a pen to pull the string taut and rotate it around the two thumbtacks. The figure that results is an ellipse.
A line drawn through the foci intersect the ellipse in two points. Each point is called a vertex of the ellipse. The segment connecting the vertices is called the major axis. The midpoint of the segment is called the center of the ellipse. A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.
We mentioned earlier that our goal is to connect the geometry of a conic with algebra. Placing the ellipse on a rectangular coordinate system gives us that opportunity. In the figure, we placed the ellipse so the foci are on the x-axis and the center is the origin.
The definition states the sum of the distance from the foci to a point is constant. So is a constant that we will call so, We will use the distance formula to lead us to an algebraic formula for an ellipse.
To graph the ellipse, it will be helpful to know the intercepts. We will find the x-intercepts and y-intercepts using the formula.
Notice that when the major axis is horizontal, the value of a will be greater than the value of b and when the major axis is vertical, the value of b will be greater than the value of a. We will use this information to graph an ellipse that is centered at the origin.
We summarize the steps for reference.
Sometimes our equation will first need to be put in standard form.
### Find the Equation of an Ellipse with Center at the Origin
If we are given the graph of an ellipse, we can find the equation of the ellipse.
### Graph an Ellipse with Center Not at the Origin
The ellipses we have looked at so far have all been centered at the origin. We will now look at ellipses whose center is
The equation is and when the major axis is horizontal so the distance from the center to the vertex is a. When the major axis is vertical so the distance from the center to the vertex is b.
If we look at the equations of and we see that they are both ellipses with and So they will have the same size and shape. They are different in that they do not have the same center.
Notice in the graph above that we could have graphed by translations. We moved the original ellipse to the right 3 units and then up 1 unit.
In the next example we will use the translation method to graph the ellipse.
When an equation has both an and a with different coefficients, we verify that it is an ellipsis by putting it in standard form. We will then be able to graph the equation.
### Solve Application with Ellipses
The orbits of the planets around the sun follow elliptical paths.
### Key Concepts
1. Ellipse: An ellipse is all points in a plane where the sum of the distances from two fixed points is constant. Each of the fixed points is called a focus of the ellipse.
If we draw a line through the foci intersects the ellipse in two points—each is called a vertex of the ellipse.
The segment connecting the vertices is called the major axis.
The midpoint of the segment is called the center of the ellipse.
A segment perpendicular to the major axis that passes through the center and intersects the ellipse in two points is called the minor axis.
2. Standard Form of the Equation an Ellipse with Center The standard form of the equation of an ellipse with center is
The x-intercepts are and
The y-intercepts are and
3. How to an Ellipse with Center
4. Standard Form of the Equation an Ellipse with Center The standard form of the equation of an ellipse with center is
When the major axis is horizontal so the distance from the center to the vertex is a.
When the major axis is vertical so the distance from the center to the vertex is b.
### Practice Makes Perfect
Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, graph each equation by translation.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
In the following exercises, graph the equation.
Solve Application with Ellipses
### 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? |
# Conics
## Hyperbolas
### Graph a Hyperbola with Center at (0, 0)
The last conic section we will look at is called a hyperbola. We will see that the equation of a hyperbola looks the same as the equation of an ellipse, except it is a difference rather than a sum. While the equations of an ellipse and a hyperbola are very similar, their graphs are very different.
We define a hyperbola as all points in a plane where the difference of their distances from two fixed points is constant. Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis. The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola. The midpoint of the segment joining the foci is called the center of the hyperbola. The line perpendicular to the transverse axis that passes through the center is called the conjugate axis. Each piece of the graph is called a branch of the hyperbola.
Again our goal is to connect the geometry of a conic with algebra. Placing the hyperbola on a rectangular coordinate system gives us that opportunity. In the figure, we placed the hyperbola so the foci are on the x-axis and the center is the origin.
The definition states the difference of the distance from the foci to a point is constant. So is a constant that we will call so We will use the distance formula to lead us to an algebraic formula for an ellipse.
To graph the hyperbola, it will be helpful to know about the intercepts. We will find the x-intercepts and y-intercepts using the formula.
The a, b values in the equation also help us find the asymptotes of the hyperbola. The asymptotes are intersecting straight lines that the branches of the graph approach but never intersect as the x, y values get larger and larger.
To find the asymptotes, we sketch a rectangle whose sides intersect the x-axis at the vertices and intersect the y-axis at The lines containing the diagonals of this rectangle are the asymptotes of the hyperbola. The rectangle and asymptotes are not part of the hyperbola, but they help us graph the hyperbola.
The asymptotes pass through the origin and we can evaluate their slope using the rectangle we sketched. They have equations and
There are two equations for hyperbolas, depending whether the transverse axis is vertical or horizontal. We can tell whether the transverse axis is horizontal by looking at the equation. When the equation is in standard form, if the x2-term is positive, the transverse axis is horizontal. When the equation is in standard form, if the y2-term is positive, the transverse axis is vertical.
The second equations could be derived similarly to what we have done. We will summarize the results here.
We will use these properties to graph hyperbolas.
We summarize the steps for reference.
Sometimes the equation for a hyperbola needs to be first placed in standard form before we graph it.
### Graph a Hyperbola with Center at
Hyperbolas are not always centered at the origin. When a hyperbola is centered at the equations changes a bit as reflected in the table.
We summarize the steps for easy reference.
Be careful as you identify the center. The standard equation has and with the center as
Again, sometimes we have to put the equation in standard form as our first step.
### Identify Conic Sections by their Equations
Now that we have completed our study of the conic sections, we will take a look at the different equations and recognize some ways to identify a conic by its equation. When we are given an equation to graph, it is helpful to identify the conic so we know what next steps to take.
To identify a conic from its equation, it is easier if we put the variable terms on one side of the equation and the constants on the other.
### Key Concepts
1. Hyperbola: A hyperbola is all points in a plane where the difference of their distances from two fixed points is constant.
Each of the fixed points is called a focus of the hyperbola.
The line through the foci, is called the transverse axis.
The two points where the transverse axis intersects the hyperbola are each a vertex of the hyperbola.
The midpoint of the segment joining the foci is called the center of the hyperbola.
The line perpendicular to the transverse axis that passes through the center is called the conjugate axis.
Each piece of the graph is called a branch of the hyperbola.
2. How to graph a hyperbola centered at
3. How to graph a hyperbola centered at
### Practice Makes Perfect
Graph a Hyperbola with Center at
In the following exercises, graph.
Graph a Hyperbola with Center at
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.
Mixed Practice
In the following exercises, graph each equation.
### 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? |
# Conics
## Solve Systems of Nonlinear Equations
### Solve a System of Nonlinear Equations Using Graphing
We learned how to solve systems of linear equations with two variables by graphing, substitution and elimination. We will be using these same methods as we look at nonlinear systems of equations with two equations and two variables. A system of nonlinear equations is a system where at least one of the equations is not linear.
For example each of the following systems is a system of nonlinear equations.
Just as with systems of linear equations, a solution of a nonlinear system is an ordered pair that makes both equations true. In a nonlinear system, there may be more than one solution. We will see this as we solve a system of nonlinear equations by graphing.
When we solved systems of linear equations, the solution of the system was the point of intersection of the two lines. With systems of nonlinear equations, the graphs may be circles, parabolas or hyperbolas and there may be several points of intersection, and so several solutions. Once you identify the graphs, visualize the different ways the graphs could intersect and so how many solutions there might be.
To solve systems of nonlinear equations by graphing, we use basically the same steps as with systems of linear equations modified slightly for nonlinear equations. The steps are listed below for reference.
To identify the graph of each equation, keep in mind the characteristics of the and terms of each conic.
### Solve a System of Nonlinear Equations Using Substitution
The graphing method works well when the points of intersection are integers and so easy to read off the graph. But more often it is difficult to read the coordinates of the points of intersection. The substitution method is an algebraic method that will work well in many situations. It works especially well when it is easy to solve one of the equations for one of the variables.
The substitution method is very similar to the substitution method that we used for systems of linear equations. The steps are listed below for reference.
So far, each system of nonlinear equations has had at least one solution. The next example will show another option.
### Solve a System of Nonlinear Equations Using Elimination
When we studied systems of linear equations, we used the method of elimination to solve the system. We can also use elimination to solve systems of nonlinear equations. It works well when the equations have both variables squared. When using elimination, we try to make the coefficients of one variable to be opposites, so when we add the equations together, that variable is eliminated.
The elimination method is very similar to the elimination method that we used for systems of linear equations. The steps are listed for reference.
There are also four options when we consider a circle and a hyperbola.
### Use a System of Nonlinear Equations to Solve Applications
Systems of nonlinear equations can be used to model and solve many applications. We will look at an everyday geometric situation as our example.
### Key Concepts
1. How to solve a system of nonlinear equations by graphing.
2. How to solve a system of nonlinear equations by substitution.
3. How to solve a system of equations by elimination.
### Section Exercises
### Practice Makes Perfect
Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.
### 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?
### Chapter Review Exercises
### Distance and Midpoint Formulas; Circles
Use the Distance Formula
In the following exercises, find the distance between the points. Round to the nearest tenth if needed.
Use the Midpoint Formula
In the following exercises, find the midpoint of the line segments whose endpoints are given.
Write the Equation of a Circle in Standard Form
In the following exercises, write the standard form of the equation of the circle with the given information.
Graph a Circle
In the following exercises, ⓐ find the center and radius, then ⓑ graph each circle.
### Parabolas
Graph Vertical Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Graph Horizontal Parabolas
In the following exercises, graph each equation by using its properties.
In the following exercises, ⓐ write the equation in standard form, then ⓑ use properties of the standard form to graph the equation.
Solve Applications with Parabolas
In the following exercises, create the equation of the parabolic arch formed in the foundation of the bridge shown. Give the answer in standard form.
### Ellipses
Graph an Ellipse with Center at the Origin
In the following exercises, graph each ellipse.
Find the Equation of an Ellipse with Center at the Origin
In the following exercises, find the equation of the ellipse shown in the graph.
Graph an Ellipse with Center Not at the Origin
In the following exercises, graph each ellipse.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Solve Applications with Ellipses
In the following exercises, write the equation of the ellipse described.
### Hyperbolas
Graph a Hyperbola with Center at
In the following exercises, graph.
Graph a Hyperbola with Center at
In the following exercises, graph.
In the following exercises, ⓐ write the equation in standard form and ⓑ graph.
Identify the Graph of each Equation as a Circle, Parabola, Ellipse, or Hyperbola
In the following exercises, identify the type of graph.
### Solve Systems of Nonlinear Equations
Solve a System of Nonlinear Equations Using Graphing
In the following exercises, solve the system of equations by using graphing.
Solve a System of Nonlinear Equations Using Substitution
In the following exercises, solve the system of equations by using substitution.
Solve a System of Nonlinear Equations Using Elimination
In the following exercises, solve the system of equations by using elimination.
Use a System of Nonlinear Equations to Solve Applications
In the following exercises, solve the problem using a system of equations.
### Practice Test
In the following exercises, find the distance between the points and the midpoint of the line segment with the given endpoints. Round to the nearest tenth as needed.
In the following exercises, write the standard form of the equation of the circle with the given information.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, and ⓑ graph the equation.
In the following exercises, ⓐ identify the type of graph of each equation as a circle, parabola, ellipse, or hyperbola, ⓑ write the equation in standard form, and ⓒ graph the equation. |
# Sequences, Series and Binomial Theorem
## Introduction
A strange charge suddenly appears on your credit card. But your card is in your wallet—it’s not even lost or stolen. Sadly, you may have been a victim of cyber crime. In this day and age, most transactions take advantage of the benefit of computers in some way. Cyber crime is any type of crime that uses a computer or computer network. Thankfully, many people are working to prevent cyber crime. Sometimes known as cryptographers, these people develop complex patterns in computer codes that block access to would-be thieves as well as write codes to intercept and decode information from them so that they may be identified. In this chapter, you will explore basic sequences and series related to those used by computer programmers to prevent cyber crime. |
# Sequences, Series and Binomial Theorem
## Sequences
### Write the First Few Terms of a Sequence
Let’s look at the function and evaluate it for just the counting numbers.
If we list the function values in order as 2, 4, 6, 8, and 10, … we have a sequence. A sequence is a function whose domain is the counting numbers.
A sequence can also be seen as an ordered list of numbers and each number in the list is a term. A sequence may have an infinite number of terms or a finite number of terms. Our sequence has three dots (ellipsis) at the end which indicates the list never ends. If the domain is the set of all counting numbers, then the sequence is an infinite sequence. Its domain is all counting numbers and there is an infinite number of counting numbers.
If we limit the domain to a finite number of counting numbers, then the sequence is a finite sequence. If we use only the first four counting numbers, 1, 2, 3, 4 our sequence would be the finite sequence,
Often when working with sequences we do not want to write out all the terms. We want more compact way to show how each term is defined. When we worked with functions, we wrote and we said the expression was the rule that defined values in the range. While a sequence is a function, we do not use the usual function notation. Instead of writing the function as we would write it as The is the nth term of the sequence, the term in the nth position where n is a value in the domain. The formula for writing the nth term of the sequence is called the general term or formula of the sequence.
When we are given the general term of the sequence, we can find the terms by replacing n with the counting numbers in order. For
To find the values of a sequence, we substitute in the counting numbers in order into the general term of the sequence.
For some sequences, the variable is an exponent.
It is not uncommon to see the expressions or in the general term for a sequence. If we evaluate each of these expressions for a few values, we see that this expression alternates the sign for the terms.
The terms in the next example will alternate signs as a result of the powers of
### Find a Formula for the General Term (nth Term) of a Sequence
Sometimes we have a few terms of a sequence and it would be helpful to know the general term or nth term. To find the general term, we look for patterns in the terms. Often the patterns involve multiples or powers. We also look for a pattern in the signs of the terms.
### Use Factorial Notation
Sequences often have terms that are products of consecutive integers. We indicate these products with a special notation called factorial notation. For example,, read 5 factorial, means The exclamation point is not punctuation here; it indicates the factorial notation.
The values of for the first 5 positive integers are shown.
When there is a fraction with factorials in the numerator and denominator, we line up the factors vertically to make our calculations easier.
### Find the Partial Sum
Sometimes in applications, rather than just list the terms, it is important for us to add the terms of a sequence. Rather than just connect the terms with plus signs, we can use summation notation.
For example, can be written as We read this as “the sum of a sub i from i equals one to five.” The symbol means to add and the i is the index of summation. The 1 tells us where to start (initial value) and the 5 tells us where to end (terminal value).
When we add a finite number of terms, we call the sum a partial sum.
The index does not always have to be i we can use any letter, but i and k are commonly used. The index does not have to start with 1 either—it can start and end with any positive integer.
### Use Summation Notation to Write a Sum
In the last two examples, we went from summation notation to writing out the sum. Now we will start with a sum and change it to summation notation. This is very similar to finding the general term of a sequence. We will need to look at the terms and find a pattern. Often the patterns involve multiples or powers.
When the terms of a sum have negative coefficients, we must carefully analyze the pattern of the signs.
### Key Concepts
1. Factorial Notation
If n is a positive integer, then is
We define as 1, so
2. Summation Notation
The sum of the first n terms of a sequence whose nth term is written in summation notation as:
The i is the index of summation and the 1 tells us where to start and the n tells us where to end.
### Practice Makes Perfect
Write the First Few Terms of a Sequence
In the following exercises, write the first five terms of the sequence whose general term is given.
Find a Formula for the General Term (
In the following exercises, find a general term for the sequence whose first five terms are shown.
Use Factorial Notation
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
Find the Partial Sum
In the following exercises, expand the partial sum and find its value.
Use Summation Notation to write a Sum
In the following exercises, write each sum using summation notation.
### 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. |
# Sequences, Series and Binomial Theorem
## Arithmetic Sequences
### Determine if a Sequence is Arithmetic
The last section introduced sequences and now we will look at two specific types of sequences that each have special properties. In this section we will look at arithmetic sequences and in the next section, geometric sequences.
An arithmetic sequence is a sequence where the difference between consecutive terms is constant. The difference between consecutive terms in an arithmetic sequence, is d, the common difference, for n greater than or equal to two.
In each of these sequences, the difference between consecutive terms is constant, and so the sequence is arithmetic.
If we know the first term, and the common difference, d, we can list a finite number of terms of the sequence.
### Find the General Term (nth Term) of an Arithmetic Sequence
Just as we found a formula for the general term of a sequence, we can also find a formula for the general term of an arithmetic sequence.
Let’s write the first few terms of a sequence where the first term is and the common difference is d. We will then look for a pattern.
As we look for a pattern we see that each term starts with .
The first term adds 0d to the , the second term adds 1d, the third term adds 2d, the fourth term adds 3d, and the fifth term adds 4d. The number of ds that were added to is one less than the number of the term. This leads us to the following
We will use this formula in the next example to find the 15th term of a sequence.
Sometimes we do not know the first term and we must use other given information to find it before we find the requested term.
Sometimes the information given leads us to two equations in two unknowns. We then use our methods for solving systems of equations to find the values needed.
### Find the Sum of the First n Terms of an Arithmetic Sequence
As with the general sequences, it is often useful to find the sum of an arithmetic sequence. The sum, of the first terms of any arithmetic sequence is written as To find the sum by merely adding all the terms can be tedious. So we can also develop a formula to find the sum of a sequence using the first and last term of the sequence.
We can develop this new formula by first writing the sum by starting with the first term, and keep adding a d to get the next term as:
We can also reverse the order of the terms and write the sum by starting with and keep subtracting d to get the next term as
If we add these two expressions for the sum of the first n terms of an arithmetic sequence, we can derive a formula for the sum of the first n terms of any arithmetic series.
Because there are n sums of on the right side of the equation, we rewrite the right side as
We divide by two to solve for
This gives us a general formula for the sum of the first n terms of an arithmetic sequence.
We apply this formula in the next example where the first few terms of the sequence are given.
In the next example, we are given the general term for the sequence and are asked to find the sum of the first 50 terms.
In the next example we are given the sum in summation notation. To add all the terms would be tedious, so we extract the information needed to use the formula to find the sum of the first n terms.
### Key Concepts
1. General Term (
The general term of an arithmetic sequence with first term and the common difference d is
2. Sum of the First
The sum, of the first n terms of an arithmetic sequence, where is the first term and is the nth term is
### Practice Makes Perfect
Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
In the following exercises, write the first five terms of each sequence with the given first term and common difference.
Find the General Term (
In the following exercises, find the term described using the information provided.
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
Find the Sum of the First
In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.
In the following exercises, find the sum of the first 50 terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.
### 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? |
# Sequences, Series and Binomial Theorem
## Geometric Sequences and Series
### Determine if a Sequence is Geometric
We are now ready to look at the second special type of sequence, the geometric sequence.
A sequence is called a geometric sequence if the ratio between consecutive terms is always the same. The ratio between consecutive terms in a geometric sequence is r, the common ratio, where n is greater than or equal to two.
Consider these sequences.
If we know the first term, and the common ratio, r, we can list a finite number of terms of the sequence.
### Find the General Term (nth Term) of a Geometric Sequence
Just as we found a formula for the general term of a sequence and an arithmetic sequence, we can also find a formula for the general term of a geometric sequence.
Let’s write the first few terms of the sequence where the first term is and the common ratio is r. We will then look for a pattern.
As we look for a pattern in the five terms above, we see that each of the terms starts with
The first term, is not multiplied by any r. In the second term, the is multiplied by r. In the third term, the is multiplied by r two times ( or ). In the fourth term, the is multiplied by r three times ( or ) and in the fifth term, the is multiplied by r four times. In each term, the number of times is multiplied by r is one less than the number of the term. This leads us to the following
We will use this formula in the next example to find the fourteenth term of a sequence.
Sometimes we do not know the common ratio and we must use the given information to find it before we find the requested term.
### Find the Sum of the First n Terms of a Geometric Sequence
We found the sum of both general sequences and arithmetic sequence. We will now do the same for geometric sequences. The sum, of the first n terms of a geometric sequence is written as We can write this sum by starting with the first term, and keep multiplying by r to get the next term as:
Let’s also multiply both sides of the equation by r.
Next, we subtract these equations. We will see that when we subtract, all but the first term of the top equation and the last term of the bottom equation subtract to zero.
We apply this formula in the next example where the first few terms of the sequence are given. Notice the sum of a geometric sequence typically gets very large when the common ratio is greater than one.
In the next example, we are given the sum in summation notation. While adding all the terms might be possible, most often it is easiest to use the formula to find the sum of the first n terms.
To use the formula, we need r. We can find it by writing out the first few terms of the sequence and find their ratio. Another option is to realize that in summation notation, a sequence is written in the form where r is the common ratio.
### Find the Sum of an Infinite Geometric Series
If we take a geometric sequence and add the terms, we have a sum that is called a geometric series. An infinite geometric series is an infinite sum whose first term is and common ratio is r and is written
We know how to find the sum of the first n terms of a geometric series using the formula, But how do we find the sum of an infinite sum?
Let’s look at the infinite geometric series Each term gets larger and larger so it makes sense that the sum of the infinite number of terms gets larger. Let’s look at a few partial sums for this series. We see and
As n gets larger and larger, the sum gets larger and larger. This is true when and we call the series divergent. We cannot find a sum of an infinite geometric series when
Let’s look at an infinite geometric series whose common ratio is a fraction less than one,. Here the terms get smaller and smaller as n gets larger. Let’s look at a few finite sums for this series. We see and
Notice the sum gets larger and larger but also gets closer and closer to one. When the expression gets smaller and smaller. In this case, we call the series convergent. As n approaches infinity, (gets infinitely large), gets closer and closer to zero. In our sum formula, we can replace the with zero and then we get a formula for the sum, S, for an infinite geometric series when
This formula gives us the sum of the infinite geometric sequence. Notice the S does not have the subscript n as in as we are not adding a finite number of terms.
An interesting use of infinite geometric series is to write a repeating decimal as a fraction.
### Apply Geometric Sequences and Series in the Real World
One application of geometric sequences has to do with consumer spending. If a tax rebate is given to each household, the effect on the economy is many times the amount of the individual rebate.
We have looked at a compound interest formula where a principal, P, is invested at an interest rate, r, for t years. The new balance, A, is when interest is compounded n times a year. This formula applies when a lump sum was invested upfront and tells us the value after a certain time period.
An annuity is an investment that is a sequence of equal periodic deposits. We will be looking at annuities that pay the interest at the time of the deposits. As we develop the formula for the value of an annuity, we are going to let That means there is one deposit per year.
Suppose P dollars is invested at the end of each year. One year later that deposit is worth dollars, and another year later it is worth dollars. After t years, it will be worth dollars.
After three years, the value of the annuity is
This a sum of the terms of a geometric sequence where the first term is P and the common ratio is We substitute these values into the sum formula. Be careful, we have two different uses of r. The r in the sum formula is the common ratio of the sequence. In this case, that is where r is the interest rate.
Remember our premise was that one deposit was made at the end of each year.
We can adapt this formula for n deposits made per year and the interest is compounded n times a year.
### Key Concepts
1. General Term ( The general term of a geometric sequence with first term and the common ratio r is
2. Sum of the First The sum, of the n terms of a geometric sequence is
where is the first term and r is the common ratio.
3. Infinite Geometric Series: An infinite geometric series is an infinite sum whose first term is and common ratio is r and is written
4. Sum of an Infinite Geometric Series: For an infinite geometric series whose first term is and common ratio r,
5. Value of an Annuity with Interest Compounded Times a Year: For a principal, P, invested at the end of a compounding period, with an interest rate, r, which is compounded n times a year, the new balance, A, after t years, is
### Practice Makes Perfect
Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
In the following exercises, determine if each sequence is arithmetic, geometric or neither. If arithmetic, indicate the common difference. If geometric, indicate the common ratio.
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
Find the General Term (
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
In the following exercises, find the indicated term of the given sequence. Find the general term for the sequence.
Find the Sum of the First
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
In the following exercises, find the sum of the geometric sequence.
Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
In the following exercises, write each repeating decimal as a fraction.
Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.
### 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? |
# Sequences, Series and Binomial Theorem
## Binomial Theorem
### Use Pascal’s Triangle to Expand a Binomial
In our previous work, we have squared binomials either by using FOIL or by using the Binomial Squares Pattern. We can also say that we expanded
To expand we recognize that this is and multiply.
To find a method that is less tedious that will work for higher expansions like we again look for patterns in some expansions.
Notice the first and last terms show only one variable. Recall that so we could rewrite the first and last terms to include both variables. For example, we could expand to show each term with both variables.
Generally, we don’t show the zero exponents, just as we usually write x rather than 1x.
Let’s look at an example to highlight the last three patterns.
From the patterns we identified, we see the variables in the expansion of would be
To find the coefficients of the terms, we write our expansions again focusing on the coefficients. We rewrite the coefficients to the right forming an array of coefficients.
The array to the right is called Pascal’s Triangle. Notice each number in the array is the sum of the two closest numbers in the row above. We can find the next row by starting and ending with one and then adding two adjacent numbers.
This triangle gives the coefficients of the terms when we expand binomials.
In the next example, we will use this triangle and the patterns we recognized to expand the binomial.
In the next example we want to expand a binomial with one variable and one constant. We need to identify the a and b to carefully apply the pattern.
In the next example, the binomial is a difference and the first term has a constant times the variable. Once we identify the a and b of the pattern, we must once again carefully apply the pattern.
### Evaluate a Binomial Coefficient
While Pascal’s Triangle is one method to expand a binomial, we will also look at another method. Before we get to that, we need to introduce some more factorial notation. This notation is not only used to expand binomials, but also in the study and use of probability.
To find the coefficients of the terms of expanded binomials, we will need to be able to evaluate the notation which is called a binomial coefficient. We read as “n choose r” or “n taken r at a time”.
In the previous example, parts (a), (b), (c) demonstrate some special properties of binomial coefficients.
### Use the Binomial Theorem to Expand a Binomial
We are now ready to use the alternate method of expanding binomials. The Binomial Theorem uses the same pattern for the variables, but uses the binomial coefficient for the coefficient of each term.
Notice that when we expanded in the last example, using the Binomial Theorem, we got the same coefficients we would get from using Pascal’s Triangle.
The next example, the binomial is a difference. When the binomial is a difference, we must be careful in identifying the values we will use in the pattern.
Things can get messy when both terms have a coefficient and a variable.
The real beauty of the Binomial Theorem is that it gives a formula for any particular term of the expansion without having to compute the whole sum. Let’s look for a pattern in the Binomial Theorem.
Notice, that in each case the exponent on the b is one less than the number of the term. The term is the term where the exponent of b is r. So we can use the format of the term to find the value of a specific term.
### Key Concepts
1. Patterns in the expansion of
2. Pascal’s Triangle
3. Binomial Coefficient
: A binomial coefficient where r and n are integers with is defined as
We read as “n choose r” or “n taken r at a time”.
4. Properties of Binomial Coefficients
5. Binomial Theorem: For any real numbers a, b, and positive integer n,
### Section Exercises
### Practice Makes Perfect
Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.
### 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?
### Chapter Review Exercises
### Sequences
Write the First Few Terms of a Sequence
In the following exercises, write the first five terms of the sequence whose general term is given.
Find a Formula for the General Term (
In the following exercises, find a general term for the sequence whose first five terms are shown.
Use Factorial Notation
In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.
Find the Partial Sum
In the following exercises, expand the partial sum and find its value.
Use Summation Notation to write a Sum
In the following exercises, write each sum using summation notation.
### Arithmetic Sequences
Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
In the following exercises, write the first five terms of each arithmetic sequence with the given first term and common difference.
Find the General Term (
In the following exercises, find the term described using the information provided.
In the following exercises, find the indicated term and give the formula for the general term.
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
Find the Sum of the First
In the following exercises, find the sum of the first 30 terms of each arithmetic sequence.
In the following exercises, find the sum of the first fifteen terms of the arithmetic sequence whose general term is given.
In the following exercises, find each sum.
### Geometric Sequences and Series
Determine if a Sequence is Geometric
In the following exercises, determine if the sequence is geometric, and if so, indicate the common ratio.
In the following exercises, write the first five terms of each geometric sequence with the given first term and common ratio.
Find the General Term (
In the following exercises, find the indicated term of a sequence where the first term and the common ratio is given.
In the following exercises, find the indicated term of the given sequence. Find the general term of the sequence.
Find the Sum of the First
In the following exercises, find the sum of the first fifteen terms of each geometric sequence.
In the following exercises, find the sum
Find the Sum of an Infinite Geometric Series
In the following exercises, find the sum of each infinite geometric series.
In the following exercises, write each repeating decimal as a fraction.
Apply Geometric Sequences and Series in the Real World
In the following exercises, solve the problem.
### Binomial Theorem
Use Pascal’s Triangle to Expand a Binomial
In the following exercises, expand each binomial using Pascal’s Triangle.
Evaluate a Binomial Coefficient
In the following exercises, evaluate.
Use the Binomial Theorem to Expand a Binomial
In the following exercises, expand each binomial, using the Binomial Theorem.
In the following exercises, find the indicated term in the expansion of the binomial.
In the following exercises, find the coefficient of the indicated term in the expansion of the binomial.
### Practice Test
In the following exercises, write the first five terms of the sequence whose general term is given.
In the following exercises, determine if the sequence is arithmetic, geometric, or neither. If arithmetic, then find the common difference. If geometric, then find the common ratio.
In the following exercises, find the sum. |
# Whole Numbers
## Introduction
Even though counting is first taught at a young age, mastering mathematics, which is the study of numbers, requires constant attention. If it has been a while since you have studied math, it can be helpful to review basic topics. In this chapter, we will focus on numbers used for counting as well as four arithmetic operations—addition, subtraction, multiplication, and division. We will also discuss some vocabulary that we will use throughout this book. |
# Whole Numbers
## Introduction to Whole Numbers
### Identify Counting Numbers and Whole Numbers
Learning algebra is similar to learning a language. You start with a basic vocabulary and then add to it as you go along. You need to practice often until the vocabulary becomes easy to you. The more you use the vocabulary, the more familiar it becomes.
Algebra uses numbers and symbols to represent words and ideas. Let’s look at the numbers first. The most basic numbers used in algebra are those we use to count objects: and so on. These are called the counting numbers. The notation “…” is called an ellipsis, which is another way to show “and so on”, or that the pattern continues endlessly. Counting numbers are also called natural numbers.
Counting numbers and whole numbers can be visualized on a number line as shown in .
The point labeled is called the origin. The points are equally spaced to the right of and labeled with the counting numbers. When a number is paired with a point, it is called the coordinate of the point.
The discovery of the number zero was a big step in the history of mathematics. Including zero with the counting numbers gives a new set of numbers called the whole numbers.
We stopped at when listing the first few counting numbers and whole numbers. We could have written more numbers if they were needed to make the patterns clear.
### Model Whole Numbers
Our number system is called a place value system because the value of a digit depends on its position, or place, in a number. The number has a different value than the number Even though they use the same digits, their value is different because of the different placement of the and the
Money gives us a familiar model of place value. Suppose a wallet contains three bills, seven bills, and four bills. The amounts are summarized in . How much money is in the wallet?
Find the total value of each kind of bill, and then add to find the total. The wallet contains
Base-10 blocks provide another way to model place value, as shown in . The blocks can be used to represent hundreds, tens, and ones. Notice that the tens rod is made up of ones, and the hundreds square is made of tens, or ones.
shows the number modeled with blocks.
### Identify the Place Value of a Digit
By looking at money and blocks, we saw that each place in a number has a different value. A place value chart is a useful way to summarize this information. The place values are separated into groups of three, called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.
Just as with the blocks, where the value of the tens rod is ten times the value of the ones block and the value of the hundreds square is ten times the tens rod, the value of each place in the place-value chart is ten times the value of the place to the right of it.
shows how the number is written in a place value chart.
1. The digit is in the millions place. Its value is
2. The digit is in the hundred thousands place. Its value is
3. The digit is in the ten thousands place. Its value is
4. The digit is in the thousands place. Its value is
5. The digit is in the hundreds place. Its value is
6. The digit is in the tens place. Its value is
7. The digit is in the ones place. Its value is
### Use Place Value to Name Whole Numbers
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 with the digit at the left, which has the largest place value. The commas separate the periods, so wherever there is a comma in the number, write a comma between the words. The ones period, which has the smallest place value, is not named.
So the number is written thirty-seven million, five hundred nineteen thousand, two hundred forty-eight.
Notice that the word and is not used when naming a whole number.
### Use Place Value to Write Whole Numbers
We will now reverse the process and write a number given in words as digits.
### Round Whole Numbers
In the U.S. Census Bureau reported the population of the state of New York as people. It might be enough to say that the population is approximately million. The word approximately means that million is not the exact population, but is close to the exact value.
The process of approximating a number is called rounding. Numbers are rounded to a specific place value depending on how much accuracy is needed. million was achieved by rounding to the millions place. Had we rounded to the one hundred thousands place, we would have as a result. Had we rounded to the ten thousands place, we would have as a result, and so on. The place value to which we round to depends on how we need to use the number.
Using the number line can help you visualize and understand the rounding process. Look at the number line in . Suppose we want to round the number to the nearest ten. Is closer to or on the number line?
Now consider the number Find in .
How do we round to the nearest ten. Find in .
So that everyone rounds the same way in cases like this, mathematicians have agreed to round to the higher number, So, rounded to the nearest ten is
Now that we have looked at this process on the number line, we can introduce a more general procedure. To round a number to a specific place, look at the number to the right of that place. If the number is less than round down. If it is greater than or equal to round up.
So, for example, to round to the nearest ten, we look at the digit in the ones place.
The digit in the ones place is a Because is greater than or equal to we increase the digit in the tens place by one. So the in the tens place becomes an Now, replace any digits to the right of the with zeros. So, rounds to
Let’s look again at rounding to the nearest Again, we look to the ones place.
The digit in the ones place is Because is less than we keep the digit in the tens place the same and replace the digits to the right of it with zero. So rounded to the nearest ten is
### Key Concepts
1. Name a whole number in words.
2. Use place value to write a whole number.
3. Round a whole number to a specific place value.
### Practice Makes Perfect
Identify Counting Numbers and Whole Numbers
In the following exercises, determine which of the following numbers are ⓐ counting numbers ⓑ whole numbers.
Model Whole Numbers
In the following exercises, use place value notation to find the value of the number modeled by the blocks.
Identify the Place Value of a Digit
In the following exercises, find the place value of the given digits.
Use Place Value to Name Whole Numbers
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers
In the following exercises, write each number as a whole number using digits.
Round Whole Numbers
In the following exercises, round to the indicated place 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. |
# Whole Numbers
## Add Whole Numbers
### Use Addition Notation
A college student has a part-time job. Last week he worked hours on Monday and hours on Friday. To find the total number of hours he worked last week, he added and
The operation of addition combines numbers to get a sum. The notation we use to find the sum of and is:
We read this as three plus four and the result is the sum of three and four. The numbers and are called the addends. A math statement that includes numbers and operations is called an expression.
### Model Addition of Whole Numbers
Addition is really just counting. We will model addition with blocks. Remember, a block represents and a rod represents Let’s start by modeling the addition expression we just considered,
Each addend is less than so we can use ones blocks.
There are blocks in all. We use an equal sign to show the sum. A math sentence that shows that two expressions are equal is called an equation. We have shown that.
When the result is or more ones blocks, we will exchange the blocks for one rod.
Next we will model adding two digit numbers.
### Add Whole Numbers Without Models
Now that we have used models to add numbers, we can move on to adding without models. Before we do that, make sure you know all the one digit addition facts. You will need to use these number facts when you add larger numbers.
Imagine filling in by adding each row number along the left side to each column number across the top. Make sure that you get each sum shown. If you have trouble, model it. It is important that you memorize any number facts you do not already know so that you can quickly and reliably use the number facts when you add larger numbers.
Did you notice what happens when you add zero to a number? The sum of any number and zero is the number itself. We call this the Identity Property of Addition. Zero is called the additive identity.
Look at the pairs of sums.
Notice that when the order of the addends is reversed, the sum does not change. This property is called the Commutative Property of Addition, which states that changing the order of the addends does not change their sum.
In the previous example, the sum of the ones and the sum of the tens were both less than But what happens if the sum is or more? Let’s use our model to find out. shows the addition of and again.
When we add the ones, we get ones. Because we have more than ones, we can exchange of the ones for ten. Now we have tens and ones. Without using the model, we show this as a small red above the digits in the tens place.
When the sum in a place value column is greater than we carry over to the next column to the left. Carrying is the same as regrouping by exchanging. For example, ones for ten or tens for hundred.
### Translate Word Phrases to Math Notation
Earlier in this section, we translated math notation into words. Now we’ll reverse the process. We’ll translate word phrases into math notation. Some of the word phrases that indicate addition are listed in .
### Add Whole Numbers in Applications
Now that we have practiced adding whole numbers, let’s use what we’ve learned to solve real-world problems. We’ll start by outlining a plan. First, we need to read the problem to determine what we are looking for. Then we write a word phrase that gives the information to find it. Next we translate the word phrase into math notation and then simplify. Finally, we write a sentence to answer the question.
Some application problems involve shapes. For example, a person might need to know the distance around a garden to put up a fence or around a picture to frame it. The perimeter is the distance around a geometric figure. The perimeter of a figure is the sum of the lengths of its sides.
### Key Concepts
1. Addition Notation To describe addition, we can use symbols and words.
2. Identity Property of Addition
3. Commutative Property of Addition
4. Add whole numbers.
### Practice Makes Perfect
Use Addition Notation
In the following exercises, translate the following from math expressions to words.
Model Addition of Whole Numbers
In the following exercises, model the addition.
Add Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, add.
Translate Word Phrases to Math Notation
In the following exercises, translate each phrase into math notation and then simplify.
Add Whole Numbers in Applications
In the following exercises, solve the problem.
In the following exercises, find the perimeter of each figure.
### 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?
|
# Whole Numbers
## Subtract Whole Numbers
### Use Subtraction Notation
Suppose there are seven bananas in a bowl. Elana uses three of them to make a smoothie. How many bananas are left in the bowl? To answer the question, we subtract three from seven. When we subtract, we take one number away from another to find the difference. The notation we use to subtract from is
We read as seven minus three and the result is the difference of seven and three.
### Model Subtraction of Whole Numbers
A model can help us visualize the process of subtraction much as it did with addition. Again, we will use blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered,
### Subtract Whole Numbers
Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.
We know because Knowing all the addition number facts will help with subtraction. Then we can check subtraction by adding. In the examples above, our subtractions can be checked by addition.
To subtract numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition. Align the digits by place value, and then subtract each column starting with the ones and then working to the left.
When we modeled subtracting from we exchanged ten for ones. When we do this without the model, we say we borrow from the tens place and add to the ones place.
### Translate Word Phrases to Math Notation
As with addition, word phrases can tell us to operate on two numbers using subtraction. To translate from a word phrase to math notation, we look for key words that indicate subtraction. Some of the words that indicate subtraction are listed in .
### Subtract Whole Numbers in Applications
To solve applications with subtraction, we will use the same plan that we used with addition. First, we need to determine what we are asked to find. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question, using the appropriate units.
### Key Concepts
1. Subtract whole numbers.
### Practice Makes Perfect
Use Subtraction Notation
In the following exercises, translate from math notation to words.
Model Subtraction of Whole Numbers
In the following exercises, model the subtraction.
Subtract Whole Numbers
In the following exercises, subtract and then check by adding.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate and simplify.
Mixed Practice
In the following exercises, simplify.
In the following exercises, translate and simplify.
Subtract Whole Numbers 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.
ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?
|
# Whole Numbers
## Multiply Whole Numbers
### Use Multiplication Notation
Suppose you were asked to count all these pennies shown in .
Would you count the pennies individually? Or would you count the number of pennies in each row and add that number times.
Multiplication is a way to represent repeated addition. So instead of adding three times, we could write a multiplication expression.
We call each number being multiplied a factor and the result the product. We read as three times eight, and the result as the product of three and eight.
There are several symbols that represent multiplication. These include the symbol as well as the dot, , and parentheses
### Model Multiplication of Whole Numbers
There are many ways to model multiplication. Unlike in the previous sections where we used blocks, here we will use counters to help us understand the meaning of multiplication. A counter is any object that can be used for counting. We will use round blue counters.
### Multiply Whole Numbers
In order to multiply without using models, you need to know all the one digit multiplication facts. Make sure you know them fluently before proceeding in this section.
shows the multiplication facts. Each box shows the product of the number down the left column and the number across the top row. If you are unsure about a product, model it. It is important that you memorize any number facts you do not already know so you will be ready to multiply larger numbers.
What happens when you multiply a number by zero? You can see that the product of any number and zero is zero. This is called the Multiplication Property of Zero.
What happens when you multiply a number by one? Multiplying a number by one does not change its value. We call this fact the Identity Property of Multiplication, and is called the multiplicative identity.
Earlier in this chapter, we learned that the Commutative Property of Addition states that changing the order of addition does not change the sum. We saw that is the same as
Is this also true for multiplication? Let’s look at a few pairs of factors.
When the order of the factors is reversed, the product does not change. This is called the Commutative Property of Multiplication.
To multiply numbers with more than one digit, it is usually easier to write the numbers vertically in columns just as we did for addition and subtraction.
We start by multiplying by
We write the in the ones place of the product. We carry the tens by writing above the tens place.
Then we multiply the by the and add the above the tens place to the product. So and Write the in the tens place of the product.
The product is
When we multiply two numbers with a different number of digits, it’s usually easier to write the smaller number on the bottom. You could write it the other way, too, but this way is easier to work with.
When we multiply by a number with two or more digits, we multiply by each of the digits separately, working from right to left. Each separate product of the digits is called a partial product. When we write partial products, we must make sure to line up the place values.
When there are three or more factors, we multiply the first two and then multiply their product by the next factor. For example:
### Translate Word Phrases to Math Notation
Earlier in this section, we translated math notation into words. Now we’ll reverse the process and translate word phrases into math notation. Some of the words that indicate multiplication are given in .
### Multiply Whole Numbers in Applications
We will use the same strategy we used previously to solve applications of multiplication. First, we need to determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify to get the answer. Finally, we write a sentence to answer the question.
If we want to know the size of a wall that needs to be painted or a floor that needs to be carpeted, we will need to find its area. The area is a measure of the amount of surface that is covered by the shape. Area is measured in square units. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm.) on a side. A square inch is a square that is one inch on each side, and so on.
For a rectangular figure, the area is the product of the length and the width. shows a rectangular rug with a length of feet and a width of feet. Each square is foot wide by foot long, or square foot. The rug is made of squares. The area of the rug is square feet.
### Key Concepts
1. Multiplication Property of Zero
2. Identity Property of Multiplication
3. Commutative Property of Multiplication
4. Multiply two whole numbers to find the product.
### Practice Makes Perfect
Use Multiplication Notation
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers
In the following exercises, model the multiplication.
Multiply Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, multiply.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Mixed Practice
In the following exercises, simplify.
In the following exercises, translate and simplify.
Multiply Whole Numbers 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?
|
# Whole Numbers
## Divide Whole Numbers
### Use Division Notation
So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the cookies in and want to package them in bags with cookies in each bag. How many bags would we need?
You might put cookies in first bag, in the second bag, and so on until you run out of cookies. Doing it this way, you would fill bags.
In other words, starting with the cookies, you would take away, or subtract, cookies at a time. Division is a way to represent repeated subtraction just as multiplication represents repeated addition.
Instead of subtracting repeatedly, we can write
We read this as twelve divided by four and the result is the quotient of and The quotient is because we can subtract from exactly times. We call the number being divided the dividend and the number dividing it the divisor. In this case, the dividend is and the divisor is
In the past you may have used the notation , but this division also can be written as In each case the is the dividend and the is the divisor.
### Model Division of Whole Numbers
As we did with multiplication, we will model division using counters. The operation of division helps us organize items into equal groups as we start with the number of items in the dividend and subtract the number in the divisor repeatedly.
### Divide Whole Numbers
We said that addition and subtraction are inverse operations because one undoes the other. Similarly, division is the inverse operation of multiplication. We know because Knowing all the multiplication number facts is very important when doing division.
We check our answer to division by multiplying the quotient by the divisor to determine if it equals the dividend. In , we know is correct because
What is the quotient when you divide a number by itself?
Dividing any number by itself produces a quotient of Also, any number divided by produces a quotient of the number. These two ideas are stated in the Division Properties of One.
Suppose we have and want to divide it among people. How much would each person get? Each person would get Zero divided by any number is
Now suppose that we want to divide by That means we would want to find a number that we multiply by to get This cannot happen because times any number is Division by zero is said to be undefined.
These two ideas make up the Division Properties of Zero.
Another way to explain why division by zero is undefined is to remember that division is really repeated subtraction. How many times can we take away from Because subtracting will never change the total, we will never get an answer. So we cannot divide a number by
When the divisor or the dividend has more than one digit, it is usually easier to use the notation. This process is called long division. Let’s work through the process by dividing by
We would repeat the process until there are no more digits in the dividend to bring down. In this problem, there are no more digits to bring down, so the division is finished.
Check by multiplying the quotient times the divisor to get the dividend. Multiply to make sure that product equals the dividend,
It does, so our answer is correct.
So far all the division problems have worked out evenly. For example, if we had cookies and wanted to make bags of cookies, we would have bags. But what if there were cookies and we wanted to make bags of Start with the cookies as shown in .
Try to put the cookies in groups of eight as in .
There are groups of eight cookies, and cookies left over. We call the cookies that are left over the remainder and show it by writing R4 next to the (The R stands for remainder.)
To check this division we multiply times to get and then add the remainder of
### Translate Word Phrases to Math Notation
Earlier in this section, we translated math notation for division into words. Now we’ll translate word phrases into math notation. Some of the words that indicate division are given in .
### Divide Whole Numbers in Applications
We will use the same strategy we used in previous sections to solve applications. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We then translate the phrase into math notation and simplify it to get the answer. Finally, we write a sentence to answer the question.
### Key Concepts
1. Division Properties of One
2. Division Properties of Zero
3. Divide whole numbers.
### Section Exercises
### Practice Makes Perfect
Use Division Notation
In the following exercises, translate from math notation to words.
Model Division of Whole Numbers
In the following exercises, model the division.
Divide Whole Numbers
In the following exercises, divide. Then check by multiplying.
Mixed Practice
In the following exercises, simplify.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications
In the following exercises, solve.
Mixed Practice
In the following exercises, solve.
### Writing Exercises
### Everyday Math
### 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
Identify Counting Numbers and Whole Numbers
In the following exercises, determine which of the following are (a) counting numbers (b) whole numbers.
Model Whole Numbers
In the following exercises, model each number using blocks and then show its value using place value notation.
Identify the Place Value of a Digit
In the following exercises, find the place value of the given digits.
Use Place Value to Name Whole Numbers
In the following exercises, name each number in words.
Use Place Value to Write Whole Numbers
In the following exercises, write as a whole number using digits.
Round Whole Numbers
In the following exercises, round to the nearest ten.
In the following exercises, round to the nearest hundred.
### Add Whole Numbers
Use Addition Notation
In the following exercises, translate the following from math notation to words.
Model Addition of Whole Numbers
In the following exercises, model the addition.
Add Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, add.
Translate Word Phrases to Math Notation
In the following exercises, translate each phrase into math notation and then simplify.
Add Whole Numbers in Applications
In the following exercises, solve.
In the following exercises, find the perimeter of each figure.
### Subtract Whole Numbers
Use Subtraction Notation
In the following exercises, translate the following from math notation to words.
Model Subtraction of Whole Numbers
In the following exercises, model the subtraction.
Subtract Whole Numbers
In the following exercises, subtract and then check by adding.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Subtract Whole Numbers in Applications
In the following exercises, solve.
### Multiply Whole Numbers
Use Multiplication Notation
In the following exercises, translate from math notation to words.
Model Multiplication of Whole Numbers
In the following exercises, model the multiplication.
Multiply Whole Numbers
In the following exercises, fill in the missing values in each chart.
In the following exercises, multiply.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Multiply Whole Numbers in Applications
In the following exercises, solve.
### Divide Whole Numbers
Use Division Notation
Translate from math notation to words.
Model Division of Whole Numbers
In the following exercises, model.
Divide Whole Numbers
In the following exercises, divide. Then check by multiplying.
Translate Word Phrases to Math Notation
In the following exercises, translate and simplify.
Divide Whole Numbers in Applications
In the following exercises, solve.
### Chapter Practice Test
Simplify.
Translate each phrase to math notation and then simplify.
In the following exercises, solve. |
# The Language of Algebra
## Introduction to the Language of Algebra
You may not realize it, but you already use algebra every day. Perhaps you figure out how much to tip a server in a restaurant. Maybe you calculate the amount of change you should get when you pay for something. It could even be when you compare batting averages of your favorite players. You can describe the algebra you use in specific words, and follow an orderly process. In this chapter, you will explore the words used to describe algebra and start on your path to solving algebraic problems easily, both in class and in your everyday life. |
# The Language of Algebra
## Use the Language of Algebra
### Use Variables and Algebraic Symbols
Greg and Alex have the same birthday, but they were born in different years. This year Greg is years old and Alex is so Alex is years older than Greg. When Greg was Alex was When Greg is Alex will be No matter what Greg’s age is, Alex’s age will always be years more, right?
In the language of algebra, we say that Greg’s age and Alex’s age are variable and the three is a constant. The ages change, or vary, so age is a variable. The years between them always stays the same, so the age difference is the constant.
In algebra, letters of the alphabet are used to represent variables. Suppose we call Greg’s age Then we could use to represent Alex’s age. See .
Letters are used to represent variables. Letters often used for variables are
To write algebraically, we need some symbols as well as numbers and variables. There are several types of symbols we will be using. In Whole Numbers, we introduced the symbols for the four basic arithmetic operations: addition, subtraction, multiplication, and division. We will summarize them here, along with words we use for the operations and the result.
In algebra, the cross symbol, is not used to show multiplication because that symbol may cause confusion. Does mean (three times ) or (three times )? To make it clear, use • or parentheses for multiplication.
We perform these operations on two numbers. When translating from symbolic form to words, or from words to symbolic form, pay attention to the words of or and to help you find the numbers.
1. The sum means add plus which we write as
2. The difference means subtract minus which we write as
3. The product means multiply times which we can write as
4. The quotient means divide by which we can write as
When two quantities have the same value, we say they are equal and connect them with an equal sign.
An inequality is used in algebra to compare two quantities that may have different values. The number line can help you understand inequalities. Remember that on the number line the numbers get larger as they go from left to right. So if we know that is greater than it means that is to the right of on the number line. We use the symbols and for inequalities.
The expressions can be read from left-to-right or right-to-left, though in English we usually read from left-to-right. In general,
When we write an inequality symbol with a line under it, such as it means or We read this is less than or equal to Also, if we put a slash through an equal sign, it means not equal.
We summarize the symbols of equality and inequality in .
Grouping symbols in algebra are much like the commas, colons, and other punctuation marks in written language. They indicate which expressions are to be kept together and separate from other expressions. lists three of the most commonly used grouping symbols in algebra.
Here are some examples of expressions that include grouping symbols. We will simplify expressions like these later in this section.
### Identify Expressions and Equations
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 a phrase. Here are some examples of expressions and how they relate to word phrases:
Notice that the 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:
### Simplify Expressions with Exponents
To simplify a numerical expression means to do all the math possible. For example, to simplify we’d first multiply to get and then add the to get 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:
Suppose we have the expression We could write this more compactly using exponential notation. Exponential notation is used in algebra to represent a quantity multiplied by itself several times. We write as and as In expressions such as the is called the base and the is called the exponent. The exponent tells us how many factors of the base we have to multiply.
We say is in exponential notation and is in expanded notation.
For powers of and we have special names.
lists some examples of expressions written in exponential notation.
To simplify an exponential expression without using a calculator, we write it in expanded form and then multiply the factors.
### Simplify Expressions Using the Order of Operations
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:
Imagine the confusion that could result if every problem had several different correct answers. The same expression should give the same result. So mathematicians established some guidelines called the order of operations, which outlines the order in which parts of an expression must be simplified.
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.
When there are multiple grouping symbols, we simplify the innermost parentheses first and work outward.
### Key Concepts
1. Equality Symbol
2. Inequality
1. Exponential Notation
Order of Operations When simplifying mathematical expressions perform the operations in the following order:
### Practice Makes Perfect
Use Variables and Algebraic Symbols
In the following exercises, translate from algebraic notation to words.
Identify Expressions and Equations
In the following exercises, determine if each is an expression or an equation.
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
Simplify Expressions Using the Order of Operations
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.
ⓑ 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. |
# The Language of Algebra
## Evaluate, Simplify, and Translate Expressions
### Evaluate Algebraic Expressions
In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.
To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.
### Identify Terms, Coefficients, and 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. Some examples of terms are
The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term is When we write the coefficient is since gives the coefficients for each of the terms in the left column.
An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.
Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?
Which of these terms are like terms?
1. The terms and are both constant terms.
2. The terms and are both terms with
3. The terms and both have
Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms
### Simplify Expressions by Combining Like Terms
We can simplify an expression by combining the like terms. What do you think would simplify to? If you thought you would be right!
We can see why this works by writing both terms as addition problems.
Add the coefficients and keep the same variable. It doesn’t matter what is. If you have of something and add more of the same thing, the result is of them. For example, oranges plus oranges is oranges. We will discuss the mathematical properties behind this later.
The expression has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms.
Now it is easier to see the like terms to be combined.
When any of the terms have negative coefficients, the procedure is the same, except that you have to subtract instead of adding to combine like terms.
### Translate Words to Algebraic Expressions
In the previous section, we listed many operation symbols that are used in algebra, and then we translated expressions and equations into word phrases and sentences. Now we’ll reverse the process and translate word phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. They are summarized in .
Look closely at these phrases using the four operations:
1. the sum of and
2. the difference of and
3. the product of and
4. the quotient of and
Each phrase tells you to operate on two numbers. Look for the words 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 to your present age? Eight more than means eight added to your present age.
How old were you seven years ago? This is seven years less than your age now. You subtract from your present age. Seven less than means seven subtracted from your present age.
Later in this course, we’ll apply our skills in algebra to solving equations. We’ll usually start by translating a word phrase to an algebraic expression. We’ll need to be clear about what the expression will represent. We’ll see how to do this in the next two examples.
### Key Concepts
1. Combine like terms.
### Practice Makes Perfect
Evaluate Algebraic Expressions
In the following exercises, evaluate the expression for the given value.
Identify Terms, Coefficients, and Like Terms
In the following exercises, list the terms in the given expression.
In the following exercises, identify the coefficient of the given term.
In the following exercises, identify all sets of like terms.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the given expression by combining like terms.
Translate English Phrases into Algebraic Expressions
In the following exercises, translate the given word phrase into an algebraic expression.
In the following exercises, write an algebraic expression.
### Everyday Math
In the following exercises, use algebraic expressions to solve the problem.
### 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?
|
# The Language of Algebra
## Solving Equations Using the Subtraction and Addition Properties of Equality
When some people hear the word algebra, they think of solving equations. The applications of solving equations are limitless and extend to all careers and fields. In this section, we will begin solving equations. We will start by solving basic equations, and then as we proceed through the course we will build up our skills to cover many different forms of equations.
### Determine Whether a Number is a Solution of an Equation
Solving an equation is like discovering the answer to a puzzle. An algebraic equation states that two algebraic expressions are equal. To solve an equation is to determine the values of the variable that make the equation a true statement. Any number that makes the equation true is called a solution of the equation. It is the answer to the puzzle!
To find the solution to an equation means to find the value of the variable that makes the equation true. Can you recognize the solution of If you said you’re right! We say is a solution to the equation because when we substitute for the resulting statement is true.
Since is a true statement, we know that is indeed a solution to the equation.
The symbol asks whether the left side of the equation is equal to the right side. Once we know, we can change to an equal sign or not-equal sign
### Model the Subtraction Property of Equality
We will use a model to help you understand how the process of solving an equation is like solving a puzzle. An envelope represents the variable – since its contents are unknown – and each counter represents one.
Suppose a desk has an imaginary line dividing it in half. We place three counters and an envelope on the left side of desk, and eight counters on the right side of the desk as in . Both sides of the desk 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 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 counters from the left side to get the envelope by itself. Those counters on the left match with on the right, so I can take them away from both sides. That leaves five counters on the right, so there must be counters in the envelope.” shows this process.
What algebraic equation is modeled by this situation? Each side of the desk represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope so the number of counters on the left side of the desk is On the right side of the desk are counters. We are told that is equal to so our equation is
Let’s write algebraically the steps we took to discover how many counters were in the envelope.
Now let’s check our solution. We substitute for in the original equation and see if we get a true statement.
Our solution is correct. Five counters in the envelope plus three more equals eight.
### Solve Equations Using the Subtraction Property of Equality
Our puzzle has given us an idea of what we need to do to solve an equation. The goal is to isolate the variable by itself on one side of the equations. In the previous examples, we used the Subtraction Property of Equality, which states that when we subtract the same quantity from both sides of an equation, we still have equality.
Think about twin brothers Andy and Bobby. They are years old. How old was Andy years ago? He was years less than so his age was or What about Bobby’s age years ago? Of course, he was also. Their ages are equal now, and subtracting the same quantity from both of them resulted in equal ages years ago.
### Solve Equations Using the Addition Property of Equality
In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to “undo” the addition in order to isolate the variable.
But suppose we have an equation with a number subtracted from the variable, such as 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, which says we can add the same number to both sides of the equation without changing the equality. Notice how it mirrors the Subtraction Property of Equality.
Remember the twins, Andy and Bobby? In ten years, Andy’s age will still equal Bobby’s age. They will both be
We can add the same number to both sides and still keep the equality.
### Translate Word Phrases to Algebraic Equations
Remember, an equation has an equal sign between two algebraic expressions. So if we have a sentence that tells us that two phrases are equal, we can translate it into an equation. We look for clue words that mean equals. Some words that translate to the equal sign are:
1. is equal to
2. is the same as
3. is
4. gives
5. was
6. will be
It may be helpful to put a box around the equals word(s) in the sentence to help you focus separately on each phrase. Then translate each phrase into an expression, and write them on each side of the equal sign.
We will practice translating word sentences into algebraic equations. Some of the sentences will be basic number facts with no variables to solve for. Some sentences will translate into equations with variables. The focus right now is just to translate the words into algebra.
### Translate to an Equation and Solve
Now let’s practice translating sentences into algebraic equations and then solving them. We will solve the equations by using the Subtraction and Addition Properties of Equality.
### Key Concepts
1. Determine whether a number is a solution to an equation.
If it is not true, the number is not a solution.
2. Subtraction Property of Equality
3. Solve an equation using the Subtraction Property of Equality.
4.
Addition Property of Equality
5. Solve an equation using the Addition Property of Equality.
### Practice Makes Perfect
Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each given value is a solution to the equation.
Model the Subtraction Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
Translate Word Phrase to Algebraic Equations
In the following exercises, translate the given sentence into an algebraic equation.
Translate to an Equation and Solve
In the following exercises, translate the given sentence into an algebraic equation and then solve it.
### 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?
|
# The Language of Algebra
## Find Multiples and Factors
### Identify Multiples of Numbers
Annie is counting the shoes in her closet. The shoes are matched in pairs, so she doesn’t have to count each one. She counts by twos: She has shoes in her closet.
The numbers are called multiples of Multiples of can be written as the product of a counting number and The first six multiples of are given below.
A multiple of a number is the product of the number and a counting number. So a multiple of would be the product of a counting number and Below are the first six multiples of
We can find the multiples of any number by continuing this process. shows the multiples of through for the first twelve counting numbers.
Recognizing the patterns for multiples of will be helpful to you as you continue in this course.
shows the counting numbers from to Multiples of are highlighted. Do you notice a pattern?
The last digit of each highlighted number in is either This is true for the product of and any counting number. So, to tell if any number is a multiple of look at the last digit. If it is then the number is a multiple of
Now let’s look at multiples of highlights all of the multiples of between and What do you notice about the multiples of
All multiples of end with either or Just like we identify multiples of by looking at the last digit, we can identify multiples of by looking at the last digit.
highlights the multiples of between and All multiples of all end with a zero.
highlights multiples of The pattern for multiples of is not as obvious as the patterns for multiples of
Unlike the other patterns we’ve examined so far, this pattern does not involve the last digit. The pattern for multiples of is based on the sum of the digits. If the sum of the digits of a number is a multiple of then the number itself is a multiple of See .
Consider the number The digits are and and their sum is Since is a multiple of we know that is also a multiple of
Look back at the charts where you highlighted the multiples of of and of Notice that the multiples of are the numbers that are multiples of both and That is because Likewise, since the multiples of are the numbers that are multiples of both and
### Use Common Divisibility Tests
Another way to say that is a multiple of is to say that is divisible by In fact, is so is Notice in that is not a multiple When we divided by we did not get a counting number, so is not divisible by
Since multiplication and division are inverse operations, the patterns of multiples that we found can be used as divisibility tests. summarizes divisibility tests for some of the counting numbers between one and ten.
### Find All the Factors of a Number
There are often several ways to talk about the same idea. So
far, we’ve seen that if is a multiple of we can say that is divisible by We know that is the product of and so we can say is a multiple of and is a multiple of We can also say is divisible by and by Another way to talk about this is to say that and are factors of When we write we can say that we have factored
In algebra, it can be useful to determine all of the factors of a number. This is called factoring a number, and it can help us solve many kinds of problems.
For example, suppose a choreographer is planning a dance for a ballet recital. There are dancers, and for a certain scene, the choreographer wants to arrange the dancers in groups of equal sizes on stage.
In how many ways can the dancers be put into groups of equal size? Answering this question is the same as identifying the factors of summarizes the different ways that the choreographer can arrange the dancers.
What patterns do you see in ? Did you notice that the number of groups times the number of dancers per group is always This makes sense, since there are always dancers.
You may notice another pattern if you look carefully at the first two columns. These two columns contain the exact same set of numbers—but in reverse order. They are mirrors of one another, and in fact, both columns list all of the factors of which are:
We can find all the factors of any counting number by systematically dividing the number by each counting number, starting with If the quotient is also a counting number, then the divisor and the quotient are factors of the number. We can stop when the quotient becomes smaller than the divisor.
### Identify Prime and Composite Numbers
Some numbers, like have many factors. Other numbers, such as have only two factors: and the number. A number with only two factors is called a prime number. A number with more than two factors is called a composite number. The number is neither prime nor composite. It has only one factor, itself.
lists the counting numbers from through along with their factors. The highlighted numbers are prime, since each has only two factors.
The prime numbers less than are There are many larger prime numbers too. In order to determine whether a number is prime or composite, we need to see if the number has any factors other than and itself. To do this, we can test each of the smaller prime numbers in order to see if it is a factor of the number. If none of the prime numbers are factors, then that number is also prime.
### Key Concepts
1. Factors If , then and are factors of , and is the product of and .
2. Find all the factors of a counting number.
3. Determine if a number is prime.
### Practice Makes Perfect
Identify Multiples of Numbers
In the following exercises, list all the multiples less than for the given number.
Use Common Divisibility Tests
In the following exercises, use the divisibility tests to determine whether each number is divisible by
Find All the Factors of a Number
In the following exercises, find all the factors of the given number.
Identify Prime and Composite Numbers
In the following exercises, determine if the given number is prime or composite.
### 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? |
# The Language of Algebra
## Prime Factorization and the Least Common Multiple
### Find the Prime Factorization of a Composite Number
In the previous section, we found the factors of a number. Prime numbers have only two factors, the number and the prime number itself. Composite numbers have more than two factors, and every composite number can be written as a unique product of primes. This is called the prime factorization of a number. When we write the prime factorization of a number, we are rewriting the number as a product of primes. Finding the prime factorization of a composite number will help you later in this course.
You may want to refer to the following list of prime numbers less than as you work through this section.
### Prime Factorization Using the Factor Tree Method
One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment—a “branch” of the factor tree.
If a factor is prime, we circle it (like a bud on a tree), and do not factor that “branch” any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree.
We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization.
For example, let’s find the prime factorization of We can start with any factor pair such as and We write and below with branches connecting them.
The factor is prime, so we circle it. The factor is composite, so we need to find its factors. Let’s use and We write these factors on the tree under the
The factor is prime, so we circle it. The factor is composite, and it factors into We write these factors under the Since is prime, we circle both
The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.
In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.
Note that we could have started our factor tree with any factor pair of We chose and but the same result would have been the same if we had started with and and
### Prime Factorization Using the Ladder Method
The ladder method is another way to find the prime factors of a composite number. It leads to the same result as the factor tree method. Some people prefer the ladder method to the factor tree method, and vice versa.
To begin building the “ladder,” divide the given number by its smallest prime factor. For example, to start the ladder for we divide by the smallest prime factor of
To add a “step” to the ladder, we continue dividing by the same prime until it no longer divides evenly.
Then we divide by the next prime; so we divide by
We continue dividing up the ladder in this way until the quotient is prime. Since the quotient, is prime, we stop here.
Do you see why the ladder method is sometimes called stacked division?
The prime factorization is the product of all the primes on the sides and top of the ladder.
Notice that the result is the same as we obtained with the factor tree method.
### Find the Least Common Multiple (LCM) of Two 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.
### Listing Multiples Method
A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of and We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.
We see that and appear in both lists. They are common multiples of and We would find more common multiples if we continued the list of multiples for each.
The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of and is
### Prime Factors Method
Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of and
We start by finding the prime factorization of each number.
Then we write each number as a product of primes, matching primes vertically when possible.
Now we bring down the primes in each column. The LCM is the product of these factors.
Notice that the prime factors of and the prime factors of are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that is the least common multiple.
### Key Concepts
1. Find the prime factorization of a composite number using the tree method.
2. Find the prime factorization of a composite number using the ladder method.
3. Find the LCM by listing multiples.
4. Find the LCM using the prime factors method.
### Section Exercises
### Practice Makes Perfect
Find the Prime Factorization of a Composite Number
In the following exercises, find the prime factorization of each number using the factor tree method.
In the following exercises, find the prime factorization of each number using the ladder method.
In the following exercises, find the prime factorization of each number using any method.
Find the Least Common Multiple (LCM) of Two Numbers
In the following exercises, find the least common multiple (LCM) by listing multiples.
In the following exercises, find the least common multiple (LCM) by using the prime factors method.
In the following exercises, find the least common multiple (LCM) using any method.
### 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
### Use the Language of Algebra
Use Variables and Algebraic Symbols
In the following exercises, translate from algebra to English.
Identify Expressions and Equations
In the following exercises, determine if each is an expression or equation.
Simplify Expressions with Exponents
In the following exercises, write in exponential form.
In the following exercises, write in expanded form.
In the following exercises, simplify each expression.
Simplify Expressions Using the Order of Operations
In the following exercises, simplify.
### Evaluate, Simplify, and Translate Expressions
Evaluate an Expression
In the following exercises, evaluate the following expressions.
Identify Terms, Coefficients and Like Terms
In the following exercises, identify the terms in each expression.
In the following exercises, identify the coefficient of each term.
In the following exercises, identify the like terms.
Simplify Expressions by Combining Like Terms
In the following exercises, simplify the following expressions by combining like terms.
Translate English Phrases to Algebraic Expressions
In the following exercises, translate the following phrases into algebraic expressions.
### Solve Equations Using the Subtraction and Addition Properties of Equality
Determine Whether a Number is a Solution of an Equation
In the following exercises, determine whether each number is a solution to the equation.
Model the Subtraction Property of Equality
In the following exercises, write the equation modeled by the envelopes and counters and then solve the equation using the subtraction property of equality.
Solve Equations using the Subtraction Property of Equality
In the following exercises, solve each equation using the subtraction property of equality.
Solve Equations using the Addition Property of Equality
In the following exercises, solve each equation using the addition property of equality.
Translate English Sentences to Algebraic Equations
In the following exercises, translate each English sentence into an algebraic equation.
Translate to an Equation and Solve
In the following exercises, translate each English sentence into an algebraic equation and then solve it.
Mixed Practice
In the following exercises, solve each equation.
### Find Multiples and Factors
Identify Multiples of Numbers
In the following exercises, list all the multiples less than for each of the following.
Use Common Divisibility Tests
In the following exercises, using the divisibility tests, determine whether each number is divisible by
Find All the Factors of a Number
In the following exercises, find all the factors of each number.
Identify Prime and Composite Numbers
In the following exercises, identify each number as prime or composite.
### Prime Factorization and the Least Common Multiple
Find the Prime Factorization of a Composite Number
In the following exercises, find the prime factorization of each number.
Find the Least Common Multiple of Two Numbers
In the following exercises, find the least common multiple of each pair of numbers.
### Everyday Math
### Chapter Practice Test
In the following exercises, translate from an algebraic equation to English phrases.
In the following exercises, identify each as an expression or equation.
In the following exercises, simplify, using the order of operations.
In the following exercises, evaluate each expression.
In the following exercises, translate each phrase into an algebraic expression.
In the following exercises, solve each equation.
In the following exercises, translate each English sentence into an algebraic equation and then solve it. |
# Integers
## Introduction to Integers
At over 29,000 feet, Mount Everest stands as the tallest peak on land. Located along the border of Nepal and China, Mount Everest is also known for its extreme climate. Near the summit, temperatures never rise above freezing. Every year, climbers from around the world brave the extreme conditions in an effort to scale the tremendous height. Only some are successful. Describing the drastic change in elevation the climbers experience and the change in temperatures requires using numbers that extend both above and below zero. In this chapter, we will describe these kinds of numbers and operations using them. |
# Integers
## Introduction to Integers
### Locate Positive and Negative Numbers on the Number Line
Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, (read as “negative one degree Fahrenheit”) is below A minus sign is shown before a number to indicate that it is negative. shows which is below
Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.
Elevations can also be represented by negative numbers. The elevation at sea level is Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about below sea level, so the elevation of the Dead Sea can be represented as See .
Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of Its position would then be as labeled in .
Both positive and negative numbers can be represented on a number line. Recall that the number line created in Add Whole Numbers started at and showed the counting numbers increasing to the right as shown in . The counting numbers on the number line are all positive. We could write a plus sign, before a positive number such as or but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.
Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with at the first mark to the left of at the next mark, and so on. See .
The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.
### Order Positive and Negative Numbers
We can use the number line to compare and order positive and negative numbers. Going from left to right, numbers increase in value. Going from right to left, numbers decrease in value. See .
Just as we did with positive numbers, we can use inequality symbols to show the ordering of positive and negative numbers. Remember that we use the notation (read is less than ) when is to the left of on the number line. We write (read is greater than ) when is to the right of on the number line. This is shown for the numbers and in .
The numbers lines to follow show a few more examples.
ⓐ
is to the right of on the number line, so
is to the left of on the number line, so
ⓑ
is to the left of on the number line, so
is to the right of on the number line, so
ⓒ
is to the right of on the number line, so
is to the left of on the number line, so
### Find Opposites
On the number line, the negative numbers are a mirror image of the positive numbers with zero in the middle. Because the numbers and are the same distance from zero, they are called opposites. The opposite of is and the opposite of is as shown in (a). Similarly, and are opposites as shown in (b).
### Opposite Notation
Just as the same word in English can have different meanings, the same symbol in algebra can have different meanings. The specific meaning becomes clear by looking at how it is used. You have seen the symbol in three different ways.
### Integers
The set of counting numbers, their opposites, and is the set of integers.
We must be very careful with the signs when evaluating the opposite of a variable.
### Simplify Expressions with Absolute Value
We saw that numbers such as and are opposites because they are the same distance from on the number line. They are both five units from The distance between and any number on the number line is called the absolute value of that number. Because distance is never negative, the absolute value of any number is never negative.
The symbol for absolute value is two vertical lines on either side of a number. So the absolute value of is written as and the absolute value of is written as as shown in .
We treat absolute value bars just like we treat parentheses in the order of operations. We simplify the expression inside first.
Absolute value bars act like grouping symbols. First simplify inside the absolute value bars as much as possible. Then take the absolute value of the resulting number, and continue with any operations outside the absolute value symbols.
### Translate Word Phrases into Expressions with Integers
Now we can translate word phrases into expressions with integers. Look for words that indicate a negative sign. For example, the word negative in “negative twenty” indicates So does the word opposite in “the opposite of
As we saw at the start of this section, negative numbers are needed to describe many real-world situations. We’ll look at some more applications of negative numbers in the next example.
### Key Concepts
1. Opposite Notation
2. Absolute Value Notation
### Practice Makes Perfect
Locate Positive and Negative Numbers on the Number Line
For the following exercises, draw a number line and locate and label the given points on that number line.
Order Positive and Negative Numbers on the Number Line
In the following exercises, order each of the following pairs of numbers, using or
Find Opposites
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 each absolute value expression.
In the following exercises, evaluate each absolute value expression.
In the following exercises, fill in to compare each expression.
In the following exercises, simplify each expression.
Translate Word Phrases into Expressions with Integers
Translate each phrase into an expression with integers. Do not simplify.
### 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. |
# Integers
## Add Integers
### Model Addition of Integers
Now that we have located positive and negative numbers on the number line, it is time to discuss arithmetic operations with 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 difficult. This difficulty relates to the way the brain learns.
The brain learns best by working with objects in the real world and then generalizing to abstract concepts. Toddlers learn quickly that if they have two cookies and their older brother steals one, they have only one left. This is a concrete example of Children learn their basic addition and subtraction facts from experiences in their everyday lives. Eventually, they know the number facts without relying on cookies.
Addition and subtraction of negative numbers have fewer real world examples that are meaningful to us. Math teachers have several different approaches, such as number lines, banking, temperatures, and so on, to make these concepts real.
We will model addition and subtraction of negatives with two color counters. We let a blue counter represent a positive and a red counter will represent a negative.
If we have one positive and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero as summarized in .
We will model four addition facts using the numbers
and are very similar. The first example adds positives and positives—both positives. The second example adds negatives and negatives—both negatives. In each case, we got a result of positives or negatives. When the signs are the same, the counters are all the same color.
Now let’s see what happens when the signs are different.
### Simplify Expressions with Integers
Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions with any integers.
For example, if you want to add you don’t have to count out blue counters and red counters.
Picture blue counters with red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because there are more negative counters.
Let’s try another one. We’ll add Imagine red counters and more red counters, so we have red counters all together. This means the sum is
Look again at the results of - .
The techniques we have used up to now extend to more complicated expressions. Remember to follow the order of operations.
### 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 when evaluating expressions.
Next we'll evaluate an expression with two variables.
### Translate Word Phrases to Algebraic Expressions
All our earlier work translating word phrases to algebra also applies to expressions that include both positive and negative numbers. Remember that the phrase the sum indicates addition.
### Add Integers in Applications
Recall that we were introduced to some situations in everyday life that use positive and negative numbers, such as temperatures, banking, and sports. For example, a debt of could be represented as Let’s practice translating and solving a few applications.
Solving applications is easy if we have a plan. First, we determine what we are looking for. Then we write a phrase that gives the information to find it. We translate the phrase into math notation and then simplify to get the answer. Finally, we write a sentence to answer the question.
### Key Concepts
1. Addition of Positive and Negative Integers
### Practice Makes Perfect
Model Addition of Integers
In the following exercises, model the expression to simplify.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Add 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.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?
|
# Integers
## Subtract Integers
### Model Subtraction of Integers
Remember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers.
We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.
Perhaps when you were younger, you read as five take away three. When we use counters, we can think of subtraction the same way.
We will model four subtraction facts using the numbers and
Notice that and are very much alike.
1. First, we subtracted positives from positives to get positives.
2. Then we subtracted negatives from negatives to get negatives.
Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.
Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value.
### Simplify Expressions with Integers
Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters.
1. Subtract
Think: We start with negative counters.
We have to subtract positives, but there are no positives to take away.
So we add neutral pairs to get the positives. Now we take away the positives.
So what’s left? We have the original negatives plus more negatives from the neutral pair. The result is negatives.
Notice, that to subtract we added negatives.
2. Subtract
Think: We start with positives.
We have to subtract negatives, but there are no negatives to take away.
So we add neutral pairs to the positives. Now we take away the negatives.
What’s left? We have the original positives plus more positives from the neutral pairs. The result is positives.
Notice that to subtract we added
While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters.
Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows:
Look at these two examples.
We see that gives the same answer as
Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract long ago. But knowing that gives the same answer as helps when we are subtracting negative numbers.
Now look what happens when we subtract a negative.
We see that gives the same result as Subtracting a negative number is like adding a positive.
Look again at the results of - .
### Evaluate Variable Expressions with Integers
Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.
### Translate Word Phrases to Algebraic Expressions
When we first introduced the operation symbols, we saw that the expression may be read in several ways as shown below.
Be careful to get and in the right order!
### Subtract Integers in 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 problem, we first need to determine what we are asked to find. Then we can 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.
Geography provides another application of negative numbers with the elevations of places below sea level.
Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!)
### Key Concepts
1. Subtraction of Integers
2. Subtraction Property
3. Solve Application Problems
### Practice Makes Perfect
Model Subtraction of Integers
In the following exercises, model each expression and simplify.
Simplify Expressions with Integers
In the following exercises, simplify each expression.
In the following exercises, simplify each expression.
Evaluate Variable Expressions with Integers
In the following exercises, evaluate each expression for the given values.
Translate Word Phrases to Algebraic Expressions
In the following exercises, translate each phrase into an algebraic expression and then simplify.
Subtract Integers in Applications
In the following exercises, solve the following applications.
### 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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