Preface

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About *Prealgebra 2e*

*Prealgebra 2e* is designed to meet scope and sequence requirements for a one-semester prealgebra course. The text introduces the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.

Coverage and Scope

*Prealgebra 2e* takes a student-support approach in its presentation of the content. The beginning, in particular, is presented as a sequence of small steps so that students gain confidence in their ability to succeed in the course. The order of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough understanding of each concept. As new ideas are presented, they are explicitly related to previous topics.

Changes to the Second Edition

The *Prealgebra 2e* revision focused on mathematical clarity and accuracy. Every Example, Try-It, Section Exercise, Review Exercise, and Practice Test item was reviewed by multiple faculty experts, and then verified by authors. This intensive effort resulted in hundreds of changes to the text, problem language, answers, instructor solutions, and graphics.

Pedagogical Foundation and Features

Additional Resources

About the Authors

Whole Numbers

Introduction

Introduction to Whole Numbers

Abstract:
By the end of this section, you will be able to:

* Identify counting numbers and whole numbers
* Model whole numbers
* Identify the place value of a digit
* Use place value to name whole numbers
* Use place value to write whole numbers
* Round 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.

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 $537$ has a different value than the number $735.$ Even though they use the same digits, their value is different because of the different placement of the $7$ and the $5.$

Identify the Place Value of a Digit

By looking at money and $\text{base-10}$ 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.

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.

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 $2013,$ the U.S. Census Bureau reported the population of the state of New York as $19,651,127$ people. It might be enough to say that the population is approximately $20$ million. The word *approximately* means that $20$ million is not the exact population, but is close to the exact value.

Add Whole Numbers

Abstract:
By the end of this section, you will be able to:

* Use addition notation
* Model addition of whole numbers
* Add whole numbers without models
* Translate word phrases to math notation
* Add whole numbers in applications

Use Addition Notation

A college student has a part-time job. Last week he worked $3$ hours on Monday and $4$ hours on Friday. To find the total number of hours he worked last week, he added $3$ and $4.$

Model Addition of Whole Numbers

Addition is really just counting. We will model addition with $\text{base-10}$ blocks. Remember, a block represents $1$ and a rod represents $10.$ Let’s start by modeling the addition expression we just considered, $3+4.$

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.

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 [link].

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.

Subtract Whole Numbers

Abstract:
By the end of this section, you will be able to:

* Use subtraction notation
* Model subtraction of whole numbers
* Subtract whole numbers
* Translate word phrases to math notation
* Subtract whole numbers in applications

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 $3$ from $7$ is

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 $\text{base-10}$ blocks. Remember a block represents 1 and a rod represents 10. Let’s start by modeling the subtraction expression we just considered, $7-3.$

Subtract Whole Numbers

Addition and subtraction are inverse operations. Addition undoes subtraction, and subtraction undoes addition.

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 [link].

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.

Multiply Whole Numbers

Abstract:
By the end of this section, you will be able to:

* Use multiplication notation
* Model multiplication of whole numbers
* Multiply whole numbers
* Translate word phrases to math notation
* Multiply whole numbers in applications

Use Multiplication Notation

Suppose you were asked to count all these pennies shown in [link].

Model Multiplication of Whole Numbers

There are many ways to model multiplication. Unlike in the previous sections where we used $\text{base-10}$ 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.

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 [link].

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.

Divide Whole Numbers

Abstract:
By the end of this section, you will be able to:

* Use division notation
* Model division of whole numbers
* Divide whole numbers
* Translate word phrases to math notation
* Divide whole numbers in applications

Use Division Notation

So far we have explored addition, subtraction, and multiplication. Now let’s consider division. Suppose you have the $12$ cookies in [link] and want to package them in bags with $4$ cookies in each bag. How many bags would we need?

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 $12÷4=3$ because $3·4=12.$ Knowing all the multiplication number facts is very important when doing division.

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 [link].

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.

The Language of Algebra

Introduction to the Language of Algebra

Use the Language of Algebra

Abstract:
By the end of this section, you will be able to:

* Use variables and algebraic symbols
* Identify expressions and equations
* Simplify expressions with exponents
* Simplify expressions using the order of operations

Use Variables and Algebraic Symbols

Greg and Alex have the same birthday, but they were born in different years. This year Greg is $20$ years old and Alex is $23,$ so Alex is $3$ years older than Greg. When Greg was $12,$ Alex was $15.$ When Greg is $35,$ Alex will be $38.$ No matter what Greg’s age is, Alex’s age will always be $3$ years more, right?

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.

Simplify Expressions with Exponents

To simplify a numerical expression means to do all the math possible. For example, to simplify $4·2+1$ we’d first multiply $4·2$ 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:

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.

Evaluate, Simplify, and Translate Expressions

Abstract:
By the end of this section, you will be able to:

* Evaluate algebraic expressions
* Identify terms, coefficients, and like terms
* Simplify expressions by combining like terms
* Translate word phrases to algebraic 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.

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 $7,y,5{x}^{2},9a,\text{and}{\rule{0.2em}{0ex}}13xy.$

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think $3x+6x$ would simplify to? If you thought $9x,$ you would be right!

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 [link].

Solving Equations Using the Subtraction and Addition Properties of Equality

Abstract:
By the end of this section, you will be able to:

* Determine whether a number is a solution of an equation
* Model the Subtraction Property of Equality
* Solve equations using the Subtraction Property of Equality
* Solve equations using the Addition Property of Equality
* Translate word phrases to algebraic equations
* Translate to an equation and solve

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!

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.

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.

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.

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:

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.

Find Multiples and Factors

Abstract:
By the end of this section, you will be able to:

* Identify multiples of numbers
* Use common divisibility tests
* Find all the factors of a number
* Identify prime and composite numbers

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: $2,4,6,8,10,12.$ She has $12$ shoes in her closet.

Use Common Divisibility Tests

Another way to say that $375$ is a multiple of $5$ is to say that $375$ is divisible by $5.$ In fact, $375÷5$ is $75,$ so $375$ is $5\cdot 75.$ Notice in [link] that $10,519$ is not a multiple $3.$ When we divided $10,519$ by $3$ we did not get a counting number, so $10,519$ is not divisible by $3.$

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 $m$ is a multiple of $n,$ we can say that $m$ is divisible by $n.$ We know that $72$ is the product of $8$ and $9,$ so we can say $72$ is a multiple of $8$ and $72$ is a multiple of $9.$ We can also say $72$ is divisible by $8$ and by $9.$ Another way to talk about this is to say that $8$ and $9$ are factors of $72.$ When we write $72=8\cdot 9$ we can say that we have factored $72.$

Identify Prime and Composite Numbers

Some numbers, like $72,$ have many factors. Other numbers, such as $7,$ have only two factors: $1$ 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 $1$ is neither prime nor composite. It has only one factor, itself.

Prime Factorization and the Least Common Multiple

Abstract:
By the end of this section, you will be able to:

* Find the prime factorization of a composite number
* Find the least common multiple (LCM) of two numbers

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 $1$ 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.

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.

Integers

Introduction to Integers

Introduction to Integers

Abstract:
By the end of this section, you will be able to:

* Locate positive and negative numbers on the number line
* Order positive and negative numbers
* Find opposites
* Simplify expressions with absolute value
* Translate word phrases to expressions with 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 $0.$ Very cold temperatures are measured in degrees below zero and can be described by negative numbers. For example, $-1\text{°F}$ (read as “negative one degree Fahrenheit”) is $1{\rule{0.2em}{0ex}}\text{degree}$ below $0.$ A minus sign is shown before a number to indicate that it is negative. [link] shows $-20\text{°F},$ which is $20{\rule{0.2em}{0ex}}\text{degrees}$ below $0.$

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 [link].

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 $2$ and $-2$ are the same distance from zero, they are called opposites. The opposite of $2$ is $-2,$ and the opposite of $-2$ is $2$ as shown in [link](a). Similarly, $3$ and $-3$ are opposites as shown in [link](b).

Simplify Expressions with Absolute Value

We saw that numbers such as $5$ and $-5$ are opposites because they are the same distance from $0$ on the number line. They are both five units from $0.$ The distance between $0$ 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.

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 $-20.$ So does the word *opposite* in “the opposite of $20\text{.”}$

Add Integers

Abstract:
By the end of this section, you will be able to:

* Model addition of integers
* Simplify expressions with integers
* Evaluate variable expressions with integers
* Translate word phrases to algebraic expressions
* Add integers in applications

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.

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.

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.

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 $\text{5}$ could be represented as $\text{−5.}$ Let’s practice translating and solving a few applications.

Subtract Integers

Abstract:
By the end of this section, you will be able to:

* Model subtraction of integers
* Simplify expressions with integers
* Evaluate variable expressions with integers
* Translate words phrases to algebraic expressions
* Subtract integers in applications

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.

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.

Now we’ll practice evaluating expressions that involve subtracting negative numbers as well as positive numbers.

When we first introduced the operation symbols, we saw that the expression $a-b$ may be read in several ways as shown below.

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.

Multiply and Divide Integers

Abstract:
By the end of this section, you will be able to:

* Multiply integers
* Divide integers
* Simplify expressions with integers
* Evaluate variable expressions with integers
* Translate word phrases to algebraic expressions

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our counter 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.

Divide Integers

Division is the inverse operation of multiplication. So, $15÷3=5$ because $5·3=15$ In words, this expression says that $\mathbf{\text{15}}$ can be divided into $\mathbf{\text{3}}$ groups of $\mathbf{\text{5}}$ each because adding five three times gives $\mathbf{\text{15}}.$ If we look at some examples of multiplying integers, we might figure out the rules for dividing integers.

Simplify Expressions with Integers

Now we’ll simplify expressions that use all four operations–addition, subtraction, multiplication, and division–with integers. Remember to follow the order of operations.

Now we can evaluate expressions that include multiplication and division with integers. Remember that to evaluate an expression, substitute the numbers in place of the variables, and then simplify.

Once again, all our prior work translating words to algebra transfers to phrases that include both multiplying and dividing integers. Remember that the key word for multiplication is *product* and for division is *quotient*.

Solve Equations Using Integers; The Division Property of Equality

Abstract:
By the end of this section, you will be able to:

* Determine whether an integer is a solution of an equation
* Solve equations with integers using the Addition and Subtraction Properties of Equality
* Model the Division Property of Equality
* Solve equations using the Division Property of Equality
* Translate to an equation and solve

In Solve Equations with the Subtraction and Addition Properties of Equality, we saw that a solution of an equation is a value of a variable that makes a true statement when substituted into that equation. In that section, we found solutions that were whole numbers. Now that we’ve worked with integers, we’ll find integer solutions to equations.

Solve Equations with Integers Using the Addition and Subtraction Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality, we solved equations similar to the two shown here using the Subtraction and Addition Properties of Equality. Now we can use them again with integers.

Model the Division Property of Equality

All of the equations we have solved so far have been of the form $x+a=b$ or $x-a=b.$ We were able to isolate the variable by adding or subtracting the constant term. Now we’ll see how to solve equations that involve division.

Solve Equations Using the Division Property of Equality

The previous examples lead to the Division Property of Equality. When you divide both sides of an equation by any nonzero number, you still have equality.

Translate to an Equation and Solve

In the past several examples, we were given an equation containing a variable. In the next few examples, we’ll have to first translate word sentences into equations with variables and then we will solve the equations.

Fractions

Introduction to Fractions

Visualize Fractions

Abstract:
By the end of this section, you will be able to:

* Understand the meaning of fractions
* Model improper fractions and mixed numbers
* Convert between improper fractions and mixed numbers
* Model equivalent fractions
* Find equivalent fractions
* Locate fractions and mixed numbers on the number line
* Order fractions and mixed numbers

Understand the Meaning of Fractions

Andy and Bobby love pizza. On Monday night, they share a pizza equally. How much of the pizza does each one get? Are you thinking that each boy gets half of the pizza? That’s right. There is one whole pizza, evenly divided into two parts, so each boy gets one of the two equal parts.

Model Improper Fractions and Mixed Numbers

In [link] (b), you had eight equal fifth pieces. You used five of them to make one whole, and you had three fifths left over. Let us use fraction notation to show what happened. You had eight pieces, each of them one fifth, $\frac{1}{5},$ so altogether you had eight fifths, which we can write as $\frac{8}{5}.$ The fraction $\frac{8}{5}$ is one whole, $1,$ plus three fifths, $\frac{3}{5},$ or $1\frac{3}{5},$ which is read as *one and three-fifths*.

Convert between Improper Fractions and Mixed Numbers

In [link], we converted the improper fraction $\frac{11}{6}$ to the mixed number $1\frac{5}{6}$ using fraction circles. We did this by grouping six sixths together to make a whole; then we looked to see how many of the $11$ pieces were left. We saw that $\frac{11}{6}$ made one whole group of six sixths plus five more sixths, showing that $\frac{11}{6}=1\frac{5}{6}.$

Model Equivalent Fractions

Let’s think about Andy and Bobby and their favorite food again. If Andy eats $\frac{1}{2}$ of a pizza and Bobby eats $\frac{2}{4}$ of the pizza, have they eaten the same amount of pizza? In other words, does $\frac{1}{2}=\frac{2}{4}?$ We can use fraction tiles to find out whether Andy and Bobby have eaten *equivalent* parts of the pizza.

Find Equivalent Fractions

We used fraction tiles to show that there are many fractions equivalent to $\frac{1}{2}.$ For example, $\frac{2}{4},\frac{3}{6},$ and $\frac{4}{8}$ are all equivalent to $\frac{1}{2}.$ When we lined up the fraction tiles, it took four of the $\frac{1}{8}$ tiles to make the same length as a $\frac{1}{2}$ tile. This showed that $\frac{4}{8}=\frac{1}{2}.$ See [link].

Locate Fractions and Mixed Numbers on the Number Line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

Order Fractions and Mixed Numbers

We can use the inequality symbols to order fractions. Remember that $a>b$ means that $a$ is to the right of $b$ on the number line. As we move from left to right on a number line, the values increase.

Multiply and Divide Fractions

Abstract:
By the end of this section, you will be able to:

* Simplify fractions
* Multiply fractions
* Find reciprocals
* Divide fractions

Simplify Fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in *simplified* form.

Multiply Fractions

A model may help you understand multiplication of fractions. We will use fraction tiles to model $\frac{1}{2}·\frac{3}{4}.$ To multiply $\frac{1}{2}$ and $\frac{3}{4},$ think $\frac{1}{2}$ of $\frac{3}{4}.$

Find Reciprocals

The fractions $\frac{2}{3}$ and $\frac{3}{2}$ are related to each other in a special way. So are $-\frac{10}{7}$ and $-\frac{7}{10}.$ Do you see how? Besides looking like upside-down versions of one another, if we were to multiply these pairs of fractions, the product would be $1.$

Divide Fractions

Why is $12÷3=4?$ We previously modeled this with counters. How many groups of $3$ counters can be made from a group of $12$ counters?

Multiply and Divide Mixed Numbers and Complex Fractions

Abstract:
By the end of this section, you will be able to:

* Multiply and divide mixed numbers
* Translate phrases to expressions with fractions
* Simplify complex fractions
* Simplify expressions written with a fraction bar

Multiply and Divide Mixed Numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number to an improper fraction. And you learned how to do that in Visualize Fractions.

Translate Phrases to Expressions with Fractions

The words *quotient* and *ratio* are often used to describe fractions. In Subtract Whole Numbers, we defined quotient as the result of division. The quotient of $a$ and $b$ is the result you get from dividing $a$ by $b,$ or $\frac{a}{b}.$ Let’s practice translating some phrases into algebraic expressions using these terms.

Simplify Complex Fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction, which is a fraction in which the numerator or the denominator contains a fraction.

Simplify Expressions with a Fraction Bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction $-\frac{1}{3}$ could be the result of dividing $\frac{-1}{3},$ a negative by a positive, or of dividing $\frac{1}{-3},$ a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

Add and Subtract Fractions with Common Denominators

Abstract:
By the end of this section, you will be able to:

* Model fraction addition
* Add fractions with a common denominator
* Model fraction subtraction
* Subtract fractions with a common denominator

Model Fraction Addition

How many quarters are pictured? One quarter plus $2$ quarters equals $3$ quarters.

Add Fractions with a Common Denominator

[link] shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into $12$ slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or $\frac{7}{12}$ of the pizza) left in the box. If Leonardo eats $2$ of these remaining pieces (or $\frac{2}{12}$ of the pizza), how much is left? There would be $5$ pieces left (or $\frac{5}{12}$ of the pizza).

Subtract Fractions with a Common Denominator

We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.

Add and Subtract Fractions with Different Denominators

Abstract:
By the end of this section, you will be able to:

* Find the least common denominator (LCD)
* Convert fractions to equivalent fractions with the LCD
* Add and subtract fractions with different denominators
* Identify and use fraction operations
* Use the order of operations to simplify complex fractions
* Evaluate variable expressions with fractions

Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of $\frac{1}{4}$ when $\frac{1}{6}$ is $12.$ We saw that three $\frac{1}{12}$ pieces exactly covered $\frac{1}{4}$ and two $\frac{1}{12}$ pieces exactly covered $\frac{1}{6},$ so

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Add and Subtract Mixed Numbers

Abstract:
By the end of this section, you will be able to:

* Model addition of mixed numbers with a common denominator
* Add mixed numbers with a common denominator
* Model subtraction of mixed numbers
* Subtract mixed numbers with a common denominator
* Add and subtract mixed numbers with different denominators

Model Addition of Mixed Numbers with a Common Denominator

So far, we’ve added and subtracted proper and improper fractions, but not mixed numbers. Let’s begin by thinking about addition of mixed numbers using money.

Add Mixed Numbers

Modeling with fraction circles helps illustrate the process for adding mixed numbers: We add the whole numbers and add the fractions, and then we simplify the result, if possible.

Model Subtraction of Mixed Numbers

Let’s think of pizzas again to model subtraction of mixed numbers with a common denominator. Suppose you just baked a whole pizza and want to give your brother half of the pizza. What do you have to do to the pizza to give him half? You have to cut it into at least two pieces. Then you can give him half.

Subtract Mixed Numbers with a Common Denominator

Now we will subtract mixed numbers without using a model. But it may help to picture the model in your mind as you read the steps.

Add and Subtract Mixed Numbers with Different Denominators

To add or subtract mixed numbers with different denominators, we first convert the fractions to equivalent fractions with the LCD. Then we can follow all the steps we used above for adding or subtracting fractions with like denominators.

Solve Equations with Fractions

Abstract:
By the end of this section, you will be able to:

* Determine whether a fraction is a solution of an equation
* Solve equations with fractions using the Addition, Subtraction, and Division Properties of Equality
* Solve equations using the Multiplication Property of Equality
* Translate sentences to equations and solve

Determine Whether a Fraction is a Solution of an Equation

As we saw in Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, a solution of an equation is a value that makes a true statement when substituted for the variable in the equation. In those sections, we found whole number and integer solutions to equations. Now that we have worked with fractions, we are ready to find fraction solutions to equations.

Solve Equations with Fractions using the Addition, Subtraction, and Division Properties of Equality

In Solve Equations with the Subtraction and Addition Properties of Equality and Solve Equations Using Integers; The Division Property of Equality, we solved equations using the Addition, Subtraction, and Division Properties of Equality. We will use these same properties to solve equations with fractions.

Solve Equations with Fractions Using the Multiplication Property of Equality

Consider the equation $\frac{x}{4}=3.$ We want to know what number divided by $4$ gives $3.$ So to “undo” the division, we will need to multiply by $4.$ The *Multiplication Property of Equality* will allow us to do this. This property says that if we start with two equal quantities and multiply both by the same number, the results are equal.

Translate Sentences to Equations and Solve

Now we have covered all four properties of equality—subtraction, addition, division, and multiplication. We’ll list them all together here for easy reference.

Decimals

Introduction to Decimals

Decimals

Abstract:
By the end of this section, you will be able to:

* Name decimals
* Write decimals
* Convert decimals to fractions or mixed numbers
* Locate decimals on the number line
* Order decimals
* Round decimals

Name Decimals

You probably already know quite a bit about decimals based on your experience with money. Suppose you buy a sandwich and a bottle of water for lunch. If the sandwich costs $\text{3.45}$, the bottle of water costs $\text{1.25}$, and the total sales tax is $\text{0.33}$, what is the total cost of your lunch?

Write Decimals

Now we will translate the name of a decimal number into decimal notation. We will reverse the procedure we just used.

Convert Decimals to Fractions or Mixed Numbers

We often need to rewrite decimals as fractions or mixed numbers. Let’s go back to our lunch order to see how we can convert decimal numbers to fractions. We know that $\text{5.03}$ means $5$ dollars and $3$ cents. Since there are $100$ cents in one dollar, $3$ cents means $\frac{3}{100}$ of a dollar, so $0.03=\frac{3}{100}.$

Locate Decimals on the Number Line

Since decimals are forms of fractions, locating decimals on the number line is similar to locating fractions on the number line.

Order Decimals

Which is larger, $0.04$ or $0.40?$

Round Decimals

In the United States, gasoline prices are usually written with the decimal part as thousandths of a dollar. For example, a gas station might post the price of unleaded gas at $\text{3.279}$ per gallon. But if you were to buy exactly one gallon of gas at this price, you would pay $\text{3.28}$, because the final price would be rounded to the nearest cent. In Whole Numbers, we saw that we round numbers to get an approximate value when the exact value is not needed. Suppose we wanted to round $\text{2.72}$ to the nearest dollar. Is it closer to $\text{2}$ or to $\text{3}?$ What if we wanted to round $\text{2.72}$ to the nearest ten cents; is it closer to $\text{2.70}$ or to $\text{2.80}?$ The number lines in [link] can help us answer those questions.

Decimal Operations

Abstract:
By the end of this section, you will be able to:

* Add and subtract decimals
* Multiply decimals
* Divide decimals
* Use decimals in money applications

Add and Subtract Decimals

Let’s take one more look at the lunch order from the start of Decimals, this time noticing how the numbers were added together.

Multiply Decimals

Multiplying decimals is very much like multiplying whole numbers—we just have to determine where to place the decimal point. The procedure for multiplying decimals will make sense if we first review multiplying fractions.

Divide Decimals

Just as with multiplication, division of decimals is very much like dividing whole numbers. We just have to figure out where the decimal point must be placed.

Use Decimals in Money Applications

We often apply decimals in real life, and most of the applications involving money. The Strategy for Applications we used in The Language of Algebra gives us a plan to follow to help find the answer. Take a moment to review that strategy now.

Decimals and Fractions

Abstract:
By the end of this section, you will be able to:

* Convert fractions to decimals
* Order decimals and fractions
* Simplify expressions using the order of operations
* Find the circumference and area of circles

Convert Fractions to Decimals

In Decimals, we learned to convert decimals to fractions. Now we will do the reverse—convert fractions to decimals. Remember that the fraction bar indicates division. So $\frac{4}{5}$ can be written $4÷5$ or $5\overline{)4}.$ This means that we can convert a fraction to a decimal by treating it as a division problem.

Order Decimals and Fractions

In Decimals, we compared two decimals and determined which was larger. To compare a decimal to a fraction, we will first convert the fraction to a decimal and then compare the decimals.

Simplify Expressions Using the Order of Operations

The order of operations introduced in Use the Language of Algebra also applies to decimals. Do you remember what the phrase “Please excuse my dear Aunt Sally” stands for?

Find the Circumference and Area of Circles

The properties of circles have been studied for over $2,000$ years. All circles have exactly the same shape, but their sizes are affected by the length of the radius, a line segment from the center to any point on the circle. A line segment that passes through a circle’s center connecting two points on the circle is called a diameter. The diameter is twice as long as the radius. See [link].

Solve Equations with Decimals

Abstract:
By the end of this section, you will be able to:

* Determine whether a decimal is a solution of an equation
* Solve equations with decimals
* Translate to an equation and solve

Determine Whether a Decimal is a Solution of an Equation

Solving equations with decimals is important in our everyday lives because money is usually written with decimals. When applications involve money, such as shopping for yourself, making your family’s budget, or planning for the future of your business, you’ll be solving equations with decimals.

Solve Equations with Decimals

In previous chapters, we solved equations using the Properties of Equality. We will use these same properties to solve equations with decimals.

Translate to an Equation and Solve

Now that we have solved equations with decimals, we are ready to translate word sentences to equations and solve. Remember to look for words and phrases that indicate the operations to use.

Averages and Probability

Abstract:
By the end of this section, you will be able to:

* Calculate the mean of a set of numbers
* Find the median of a set of numbers
* Find the mode of a set of numbers
* Apply the basic definition of probability

Calculate the Mean of a Set of Numbers

The mean is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Find the Median of a Set of Numbers

When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in [link].

Identify the Mode of a Set of Numbers

The *average* is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average. Yes, it can be confusing when the word average refers to two different numbers, the mean and the median! In fact, there is a third number that is also an average. This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.

Use the Basic Definition of Probability

The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.

Ratios and Rate

Abstract:
By the end of this section, you will be able to:

* Write a ratio as a fraction
* Write a rate as a fraction
* Find unit rates
* Find unit price
* Translate phrases to expressions with fractions

Write a Ratio as a Fraction

When you apply for a mortgage, the loan officer will compare your total debt to your total income to decide if you qualify for the loan. This comparison is called the debt-to-income ratio. A ratio compares two quantities that are measured with the same unit. If we compare $a$ and $b$, the ratio is written as $a{\rule{0.2em}{0ex}}\text{to}{\rule{0.2em}{0ex}}b,{\rule{0.2em}{0ex}}\frac{a}{b},{\rule{0.2em}{0ex}}\text{or}{\rule{0.2em}{0ex}}\mathit{\text{a}}\text{:}\mathit{\text{b}}\text{.}$

Write a Rate as a Fraction

Frequently we want to compare two different types of measurements, such as miles to gallons. To make this comparison, we use a rate. Examples of rates are $120$ miles in $2$ hours, $160$ words in $4$ minutes, and $\text{5}$ dollars per $64$ ounces.

Find Unit Rates

In the last example, we calculated that Bob was driving at a rate of $\frac{\text{175 miles}}{\text{3 hours}}.$ This tells us that every three hours, Bob will travel $175$ miles. This is correct, but not very useful. We usually want the rate to reflect the number of miles in one hour. A rate that has a denominator of $1$ unit is referred to as a unit rate.

Find Unit Price

Sometimes we buy common household items ‘in bulk’, where several items are packaged together and sold for one price. To compare the prices of different sized packages, we need to find the unit price. To find the unit price, divide the total price by the number of items. A unit price is a unit rate for one item.

Translate Phrases to Expressions with Fractions

Have you noticed that the examples in this section used the comparison words *ratio of, to, per, in, for, on*, and *from*? When you translate phrases that include these words, you should think either ratio or rate. If the units measure the same quantity (length, time, etc.), you have a ratio. If the units are different, you have a rate. In both cases, you write a fraction.

Simplify and Use Square Roots

Abstract:
By the end of this section, you will be able to:

* Simplify expressions with square roots
* Estimate square roots
* Approximate square roots
* Simplify variable expressions with square roots
* Use square roots in applications

Simplify Expressions with Square Roots

To start this section, we need to review some important vocabulary and notation.

Estimate Square Roots

So far we have only worked with square roots of perfect squares. The square roots of other numbers are not whole numbers.

Approximate Square Roots with a Calculator

There are mathematical methods to approximate square roots, but it is much more convenient to use a calculator to find square roots. Find the $\sqrt{}$ or $\sqrt{x}$ key on your calculator. You will to use this key to approximate square roots. When you use your calculator to find the square root of a number that is not a perfect square, the answer that you see is not the exact number. It is an approximation, to the number of digits shown on your calculator’s display. The symbol for an approximation is $\approx$ and it is read *approximately*.

Simplify Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

Percents

Introduction to Percents

Understand Percent

Abstract:
By the end of this section, you will be able to:

* Use the definition of percent
* Convert percents to fractions and decimals
* Convert decimals and fractions to percents

Use the Definition of Percent

How many cents are in one dollar? There are $100$ cents in a dollar. How many years are in a century? There are $100$ years in a century. Does this give you a clue about what the word “percent” means? It is really two words, “per cent,” and means per one hundred. A percent is a ratio whose denominator is $100.$ We use the percent symbol $\text{%,}$ to show percent.

Convert Percents to Fractions and Decimals

Since percents are ratios, they can easily be expressed as fractions. Remember that percent means per $100,$ so the denominator of the fraction is $100.$

Convert Decimals and Fractions to Percents

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.

Solve General Applications of Percent

Abstract:
By the end of this section, you will be able to:

* Translate and solve basic percent equations
* Solve applications of percent
* Find percent increase and percent decrease

Translate and Solve Basic Percent Equations

We will solve percent equations by using the methods we used to solve equations with fractions or decimals. In the past, you may have solved percent problems by setting them up as proportions. That was the best method available when you did not have the tools of algebra. Now as a prealgebra student, you can translate word sentences into algebraic equations, and then solve the equations.

Solve Applications of Percent

Many applications of percent occur in our daily lives, such as tips, sales tax, discount, and interest. To solve these applications we'll translate to a basic percent equation, just like those we solved in the previous examples in this section. Once you translate the sentence into a percent equation, you know how to solve it.

Find Percent Increase and Percent Decrease

People in the media often talk about how much an amount has increased or decreased over a certain period of time. They usually express this increase or decrease as a percent.

Solve Sales Tax, Commission, and Discount Applications

Abstract:
By the end of this section, you will be able to:

* Solve sales tax applications
* Solve commission applications
* Solve discount applications
* Solve mark-up applications

Solve Sales Tax Applications

Sales tax and commissions are applications of percent in our everyday lives. To solve these applications, we will follow the same strategy we used in the section on decimal operations. We show it again here for easy reference.

Solve Commission Applications

Sales people often receive a commission, or percent of total sales, for their sales. Their income may be just the commission they earn, or it may be their commission added to their hourly wages or salary. The commission they earn is calculated as a certain percent of the price of each item they sell. That percent is called the **rate of commission**.

Solve Discount Applications

Applications of discount are very common in retail settings [link]. When you buy an item on sale, the **original price** of the item has been reduced by some dollar amount. The **discount rate**, usually given as a percent, is used to determine the amount of the discount. To determine the **amount of discount**, we multiply the discount rate by the original price. We summarize the discount model in the box below.

Solve Mark-up Applications

Applications of mark-up are very common in retail settings. The price a retailer pays for an item is called the **wholesale price**. The retailer then adds a **mark-up** to the wholesale price to get the **list price**, the price he sells the item for. The mark-up is usually calculated as a percent of the wholesale price. The percent is called the **mark-up rate**. To determine the amount of mark-up, multiply the mark-up rate by the wholesale price. We summarize the mark-up model in the box below.

Solve Simple Interest Applications

Abstract:
By the end of this section, you will be able to:

* Use the simple interest formula
* Solve simple interest applications

Use the Simple Interest Formula

Do you know that banks pay you to let them keep your money? The money you put in the bank is called the **principal**, $P,$ and the bank pays you **interest**, $I.$ 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 for time, $t,$ represents the number of years the money is left in the account.

Solve Simple Interest Applications

Applications with simple interest usually involve either investing money or borrowing money. To solve these applications, we continue to use the same strategy for applications that we have used earlier in this chapter. The only difference is that in place of translating to get an equation, we can use the simple interest formula.

Solve Proportions and their Applications

Abstract:
By the end of this section, you will be able to:

* Use the definition of proportion
* Solve proportions
* Solve applications using proportions
* Write percent equations as proportions
* Translate and solve percent proportions

Use the Definition of Proportion

In the section on Ratios and Rates we saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.

Solve Proportions

To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.

Solve Applications Using Proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

Write Percent Equations As Proportions

Previously, we solved percent equations by applying the properties of equality we have used to solve equations throughout this text. Some people prefer to solve percent equations by using the proportion method. The proportion method for solving percent problems involves a percent proportion. A **percent proportion** is an equation where a percent is equal to an equivalent ratio.

Translate and Solve Percent Proportions

Now that we have written percent equations as proportions, we are ready to solve the equations.

The Properties of Real Numbers

Introduction to the Properties of Real Numbers

Rational and Irrational Numbers

Abstract:
By the end of this section, you will be able to:

* Identify rational numbers and irrational numbers
* Classify different types of real numbers

Identify Rational Numbers and Irrational Numbers

Congratulations! You have completed the first six chapters of this book! It's time to take stock of what you have done so far in this course and think about what is ahead. You have learned how to add, subtract, multiply, and divide whole numbers, fractions, integers, and decimals. You have become familiar with the language and symbols of algebra, and have simplified and evaluated algebraic expressions. You have solved many different types of applications. You have established a good solid foundation that you need so you can be successful in algebra.

Classify Real Numbers

We have seen that all counting numbers are whole numbers, all whole numbers are integers, and all integers are rational numbers. Irrational numbers are a separate category of their own. When we put together the rational numbers and the irrational numbers, we get the set of real numbers.

Commutative and Associative Properties

Abstract:
By the end of this section, you will be able to:

* Use the commutative and associative properties
* Evaluate expressions using the commutative and associative properties
* Simplify expressions using the commutative and associative properties

Use the Commutative and Associative Properties

Think about adding two numbers, such as $5$ and $3.$

Evaluate Expressions using the Commutative and Associative Properties

The commutative and associative properties can make it easier to evaluate some algebraic expressions. Since order does not matter when adding or multiplying three or more terms, we can rearrange and re-group terms to make our work easier, as the next several examples illustrate.

Simplify Expressions Using the Commutative and Associative Properties

When we have to simplify algebraic expressions, we can often make the work easier by applying the Commutative or Associative Property first instead of automatically following the order of operations. Notice that in [link] part ⓑ was easier to simplify than part ⓐ because the opposites were next to each other and their sum is $0.$ Likewise, part ⓑ in [link] was easier, with the reciprocals grouped together, because their product is $1.$ In the next few examples, we’ll use our number sense to look for ways to apply these properties to make our work easier.

Distributive Property

Abstract:
By the end of this section, you will be able to:

* Simplify expressions using the distributive property
* Evaluate expressions using the distributive property

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $\text{9.25};$ that is, $9$ dollars and $1$ quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

Properties of Identity, Inverses, and Zero

Abstract:
By the end of this section, you will be able to:

* Recognize the identity properties of addition and multiplication
* Use the inverse properties of addition and multiplication
* Use the properties of zero
* Simplify expressions using the properties of identities, inverses, and zero

Recognize the Identity Properties of Addition and Multiplication

What happens when we add zero to any number? Adding zero doesn’t change the value. For this reason, we call $0$ the additive identity.

Use the Inverse Properties of Addition and Multiplication

Notice that in each case, the missing number was the opposite of the number.

Use the Properties of Zero

We have already learned that zero is the additive identity, since it can be added to any number without changing the number’s identity. But zero also has some special properties when it comes to multiplication and division.

Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

Systems of Measurement

Abstract:
By the end of this section, you will be able to:

* Make unit conversions in the U.S. system
* Use mixed units of measurement in the U.S. system
* Make unit conversions in the metric system
* Use mixed units of measurement in the metric system
* Convert between the U.S. and the metric systems of measurement
* Convert between Fahrenheit and Celsius temperatures

Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.

Use Mixed Units of Measurement in the U.S. System

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of $10.$ The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is $1000$ meters; the prefix *kilo-* means thousand. One centimeter is $\frac{1}{100}$ of a meter, because the prefix *centi-* means one one-hundredth (just like one cent is $\frac{1}{100}$ of one dollar).

Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the U.S. system. But it may be easier because of the relation of the units to the powers of $10.$ We still must make sure to add or subtract like units.

Convert Between U.S. and Metric Systems of Measurement

Many measurements in the United States are made in metric units. A drink may come in $\text{2-liter}$ bottles, calcium may come in $\text{500-mg}$ capsules, and we may run a $\text{5-K}$ race. To work easily in both systems, we need to be able to convert between the two systems.

Convert Between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for $22\text{°C}.$ What does that mean?

Solving Linear Equations

Introduction to Solving Linear Equations

Solve Equations Using the Subtraction and Addition Properties of Equality

Abstract:
By the end of this section, you will be able to:

* Solve equations using the Subtraction and Addition Properties of Equality
* Solve equations that need to be simplified
* Translate an equation and solve
* Translate and solve applications

We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.

Solve Equations That Need to Be Simplified

In the examples up to this point, we have been able to isolate the variable with just one operation. Many of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality. You should always simplify as much as possible before trying to isolate the variable.

Translate an Equation and Solve

In previous chapters, we translated word sentences into equations. The first step is to look for the word (or words) that translate(s) to the equal sign. [link] reminds us of some of the words that translate to the equal sign.

Translate and Solve Applications

In most of the application problems we solved earlier, we were able to find the quantity we were looking for by simplifying an algebraic expression. Now we will be using equations to solve application problems. We’ll start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for.

Solve Equations Using the Division and Multiplication Properties of Equality

Abstract:
By the end of this section, you will be able to:

* Solve equations using the Division and Multiplication Properties of Equality
* Solve equations that need to be simplified

We introduced the Multiplication and Division Properties of Equality in Solve Equations Using Integers; The Division Property of Equality and Solve Equations with Fractions. We modeled how these properties worked using envelopes and counters and then applied them to solving equations (See Solve Equations Using Integers; The Division Property of Equality). We restate them again here as we prepare to use these properties again.

Solve Equations That Need to be Simplified

Many equations start out more complicated than the ones we’ve just solved. First, we need to simplify both sides of the equation as much as possible

Solve Equations with Variables and Constants on Both Sides

Abstract:
By the end of this section, you will be able to:

* Solve an equation with constants on both sides
* Solve an equation with variables on both sides
* Solve an equation with variables and constants on both sides
* Solve equations using a general strategy

Solve an Equation with Constants on Both Sides

You may have noticed that in all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we’ll see how to solve equations where the variable terms and/or constant terms are on both sides of the equation.

Solve an Equation with Variables on Both Sides

What if there are variables on both sides of the equation? We will start like we did above—choosing a variable side and a constant side, and then use the Subtraction and Addition Properties of Equality to collect all variables on one side and all constants on the other side. Remember, what you do to the left side of the equation, you must do to the right side too.

The next example will be the first to have variables *and* constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side.

Solve Equations Using a General Strategy

Each of the first few sections of this chapter has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve *any* linear equation. We call this the *general strategy*. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

Solve Equations with Fraction or Decimal Coefficients

Abstract:
By the end of this section, you will be able to:

* Solve equations with fraction coefficients
* Solve equations with decimal coefficients

Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation $\frac{1}{8}{\rule{0.1em}{0ex}}x+\frac{1}{2}=\frac{1}{4}.$

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money and percent. But decimals are really another way to represent fractions. For example, $0.3=\frac{3}{10}$ and $0.17=\frac{17}{100}.$ So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Math Models and Geometry

Introduction

Use a Problem Solving Strategy

Abstract:
By the end of this section, you will be able to:

* Approach word problems with a positive attitude
* Use a problem solving strategy for word problems
* Solve number problems

Approach Word Problems with a Positive Attitude

The world is full of word problems. How much money do I need to fill the car with gas? How much should I tip the server at a restaurant? How many socks should I pack for vacation? How big a turkey do I need to buy for Thanksgiving dinner, and what time do I need to put it in the oven? If my sister and I buy our mother a present, how much will each of us pay?

Use a Problem-solving Strategy for Word Problems

In earlier chapters, you translated word phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. Since then you've increased your math vocabulary as you learned about more algebraic procedures, and you've had more practice translating from words into algebra.

Solve Number Problems

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem Solving Strategy. Remember to look for clue words such as *difference*, *of*, and *and*.

Solve Money Applications

Abstract:
By the end of this section, you will be able to:

* Solve coin word problems
* Solve ticket and stamp word problems

Solve Coin Word Problems

Imagine taking a handful of coins from your pocket or purse and placing them on your desk. How would you determine the value of that pile of coins?

Solve Ticket and Stamp Word Problems

The strategies we used for coin problems can be easily applied to some other kinds of problems too. Problems involving tickets or stamps are very similar to coin problems, for example. Like coins, tickets and stamps have different values; so we can organize the information in tables much like we did for coin problems.

Use Properties of Angles, Triangles, and the Pythagorean Theorem

Abstract:
By the end of this section, you will be able to:

* Use the properties of angles
* Use the properties of triangles
* Use the Pythagorean Theorem

Use the Properties of Angles

Are you familiar with the phrase ‘do a $180\text{’?}$ It means to turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is $180$ degrees. See [link].

Use the Properties of Triangles

What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The triangle in [link] is called $\text{Δ}ABC,$ read ‘triangle $\text{ABC}$’. We label each side with a lower case letter to match the upper case letter of the opposite vertex.

Use the Pythagorean Theorem

The Pythagorean Theorem is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around $500$ BCE.

Use Properties of Rectangles, Triangles, and Trapezoids

Abstract:
By the end of this section, you will be able to:

* Understand linear, square, and cubic measure
* Use properties of rectangles
* Use properties of triangles
* Use properties of trapezoids

Understand Linear, Square, and Cubic Measure

When you measure your height or the length of a garden hose, you use a ruler or tape measure ([link]). A tape measure might remind you of a line—you use it for linear measure, which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure.

Use the Properties of Rectangles

A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, $L,$ and the adjacent side as the width, $W.$ See [link].

Use the Properties of Triangles

We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in [link], we’ve labeled the length $b$ and the width $h,$ so it’s area is $bh.$

Use the Properties of Trapezoids

A trapezoid is four-sided figure, a *quadrilateral*, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base $b,$ and the length of the bigger base $B.$ The height, $h,$ of a trapezoid is the distance between the two bases as shown in [link].

Solve Geometry Applications: Circles and Irregular Figures

Abstract:
By the end of this section, you will be able to:

* Use the properties of circles
* Find the area of irregular figures

Use the Properties of Circles

Do you remember the properties of circles from Decimals and Fractions Together? We’ll show them here again to refer to as we use them to solve applications.

Find the Area of Irregular Figures

So far, we have found area for rectangles, triangles, trapezoids, and circles. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.

Solve Geometry Applications: Volume and Surface Area

Abstract:
By the end of this section, you will be able to:

* Find volume and surface area of rectangular solids
* Find volume and surface area of spheres
* Find volume and surface area of cylinders
* Find volume of cones

Find Volume and Surface Area of Rectangular Solids

A cheerleading coach is having the squad paint wooden crates with the school colors to stand on at the games. (See [link]). The amount of paint needed to cover the outside of each box is the surface area, a square measure of the total area of all the sides. The amount of space inside the crate is the volume, a cubic measure.

Find the Volume and Surface Area of Spheres

A sphere is the shape of a basketball, like a three-dimensional circle. Just like a circle, the size of a sphere is determined by its radius, which is the distance from the center of the sphere to any point on its surface. The formulas for the volume and surface area of a sphere are given below.

Find the Volume and Surface Area of a Cylinder

If you have ever seen a can of soda, you know what a cylinder looks like. A cylinder is a solid figure with two parallel circles of the same size at the top and bottom. The top and bottom of a cylinder are called the bases. The height $h$ of a cylinder is the distance between the two bases. For all the cylinders we will work with here, the sides and the height, $h$ , will be perpendicular to the bases.

Find the Volume of Cones

The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.

Summary of Geometry Formulas

The following charts summarize all of the formulas covered in this chapter.

Solve a Formula for a Specific Variable

Abstract:
By the end of this section, you will be able to:

* Use the distance, rate, and time formula
* Solve a formula for a specific variable

Use the Distance, Rate, and Time Formula

One formula you’ll use often in algebra and in everyday life is the formula for distance traveled by an object moving at a constant speed. The basic idea is probably already familiar to you. Do you know what distance you travel if you drove at a steady rate of $60$ miles per hour for $2$ hours? (This might happen if you use your car’s cruise control while driving on the Interstate.) If you said $120$ miles, you already know how to use this formula!

Solve a Formula for a Specific Variable

In this chapter, you became familiar with some formulas used in geometry. Formulas are also very useful in the sciences and social sciences—fields such as chemistry, physics, biology, psychology, sociology, and criminal justice. Healthcare workers use formulas, too, even for something as routine as dispensing medicine. The widely used spreadsheet program Microsoft ExcelTM relies on formulas to do its calculations. Many teachers use spreadsheets to apply formulas to compute student grades. It is important to be familiar with formulas and be able to manipulate them easily.

Polynomials

Introduction to Polynomials

Add and Subtract Polynomials

Abstract:
By the end of this section, you will be able to:

* Identify polynomials, monomials, binomials, and trinomials
* Determine the degree of polynomials
* Add and subtract monomials
* Add and subtract polynomials
* Evaluate a polynomial for a given value

Identify Polynomials, Monomials, Binomials, and Trinomials

In Evaluate, Simplify, and Translate Expressions, you learned that a term is a constant or the product of a constant and one or more variables. When it is of the form $a{x}^{m},$ where $a$ is a constant and $m$ is a whole number, it is called a monomial. A monomial, or a sum and/or difference of monomials, is called a polynomial.

Determine the Degree of Polynomials

In this section, we will work with polynomials that have only one variable in each term. The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

Add and Subtract Monomials

In The Language of Algebra, you simplified expressions by combining like terms. Adding and subtracting monomials is the same as combining like terms. Like terms must have the same variable with the same exponent. Recall that when combining like terms only the coefficients are combined, never the exponents.

Add and Subtract Polynomials

Adding and subtracting polynomials can be thought of as just adding and subtracting like terms. Look for like terms—those with the same variables with the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together. It may also be helpful to underline, circle, or box like terms.

Evaluate a Polynomial for a Given Value

In The Language of Algebra we evaluated expressions. Since polynomials are expressions, we'll follow the same procedures to evaluate polynomials—substitute the given value for the variable into the polynomial, and then simplify.

Use Multiplication Properties of Exponents

Abstract:
By the end of this section, you will be able to:

* Simplify expressions with exponents
* Simplify expressions using the Product Property of Exponents
* Simplify expressions using the Power Property of Exponents
* Simplify expressions using the Product to a Power Property
* Simplify expressions by applying several properties
* Multiply monomials

Simplify Expressions with Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, ${2}^{4}$ means to multiply four factors of $2,$ so ${2}^{4}$ means $2·2·2·2.$ This format is known as exponential notation.

Simplify Expressions Using the Product Property of Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too. We’ll derive the properties of exponents by looking for patterns in several examples. All the exponent properties hold true for any real numbers, but right now we will only use whole number exponents.

Simplify Expressions Using the Power Property of 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.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Look for a pattern.

Simplify Expressions by Applying Several Properties

We now have three properties for multiplying expressions with exponents. Let’s summarize them and then we’ll do some examples that use more than one of the properties.

Multiply Monomials

Since a monomial is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Multiply Polynomials

Abstract:
By the end of this section, you will be able to:

* Multiply a polynomial by a monomial
* Multiply a binomial by a binomial
* Multiply a trinomial by a binomial

Multiply a Polynomial by a Monomial

In Distributive Property you learned to use the Distributive Property to simplify expressions such as $2\left(x-3\right).$ You multiplied both terms in the parentheses, $x{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}3,$ by $2,$ to get $2x-6.$ With this chapter's new vocabulary, you can say you were multiplying a binomial, $x-3,$ by a monomial, $2.$ Multiplying a binomial by a monomial is nothing new for you!

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.

Multiply a Trinomial by a Binomial

We have multiplied monomials by monomials, monomials by polynomials, and binomials by binomials. Now we're ready to multiply a trinomial by a binomial. Remember, the FOIL method will not work in this case, but we can use either the Distributive Property or the Vertical Method. We first look at an example using the Distributive Property.

Divide Monomials

Abstract:
By the end of this section, you will be able to:

* Simplify expressions using the Quotient Property of Exponents
* Simplify expressions with zero exponents
* Simplify expressions using the Quotient to a Power Property
* Simplify expressions by applying several properties
* Divide monomials

Simplify Expressions Using the Quotient Property of Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties here.

Simplify Expressions with Zero Exponents

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\frac{{a}^{m}}{{a}^{m}}.$ From earlier work with fractions, we know that

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

We'll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.

Integer Exponents and Scientific Notation

Abstract:
By the end of this section, you will be able to:

* Use the definition of a negative exponent
* Simplify expressions with integer exponents
* Convert from decimal notation to scientific notation
* Convert scientific notation to decimal form
* Multiply and divide using scientific notation

Use the Definition of a Negative Exponent

The Quotient Property of Exponents, introduced in Divide Monomials, had two forms depending on whether the exponent in the numerator or denominator was larger.

Simplify Expressions with Integer Exponents

All the exponent properties we developed earlier in this chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of $10.$ We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on.

Convert Scientific Notation to Decimal Form

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

Multiply and Divide Using Scientific Notation

We use the Properties of Exponents to multiply and divide numbers in scientific notation.

Introduction to Factoring Polynomials

Abstract:
By the end of this section, you will be able to:

* Find the greatest common factor of two or more expressions
* Factor the greatest common factor from a polynomial

Find the Greatest Common Factor of Two or More Expressions

Earlier we multiplied factors together to get a product. Now, we will be reversing this process; we will start with a product and then break it down into its factors. Splitting a product into factors is called factoring.

Factor the Greatest Common Factor from a Polynomial

Just like in arithmetic, where it is sometimes useful to represent a number in factored form (for example, $12$ as $2·6{\rule{0.2em}{0ex}}\text{or}{\rule{0.2em}{0ex}}3·4\text{),}$ in algebra it can be useful to represent a polynomial in factored form. One way to do this is by finding the greatest common factor of all the terms. Remember that you can multiply a polynomial by a monomial as follows:

Graphs

Graphs

Use the Rectangular Coordinate System

Abstract:
By the end of this section, you will be able to:

* Plot points on a rectangular coordinate system
* Identify points on a graph
* Verify solutions to an equation in two variables
* Complete a table of solutions to a linear equation
* Find solutions to linear equations in two variables

Plot Points on a Rectangular Coordinate System

Many maps, such as the Campus Map shown in [link], use a grid system to identify locations. Do you see the numbers $1,2,3,$ and $4$ across the top and bottom of the map and the letters A, B, C, and D along the sides? Every location on the map can be identified by a number and a letter.

Identify Points on a Graph

In algebra, being able to identify the coordinates of a point shown on a graph is just as important as being able to plot points. To identify the *x*-coordinate of a point on a graph, read the number on the *x*-axis directly above or below the point. To identify the *y*-coordinate of a point, read the number on the *y*-axis directly to the left or right of the point. Remember, to write the ordered pair using the correct order $\left(x,y\right).$

Verify Solutions to an Equation in Two Variables

All the equations we solved so far have been equations with one variable. In almost every case, when we solved the equation we got exactly one solution. The process of solving an equation ended with a statement such as $x=4.$ Then we checked the solution by substituting back into the equation.

Complete a Table of Solutions to a Linear Equation

In the previous examples, we substituted the $x\text{- and}{\rule{0.2em}{0ex}}y\text{-values}$ of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do we find the ordered pairs if they are not given? One way is to choose a value for $x$ and then solve the equation for $y.$ Or, choose a value for $y$ and then solve for $x.$

Find Solutions to Linear Equations in Two Variables

To find a solution to a linear equation, we can choose any number we want to substitute into the equation for either $x$ or $y.$ We could choose $1,100,1,000,$ or any other value we want. But it’s a good idea to choose a number that’s easy to work with. We’ll usually choose $0$ as one of our values.

Graphing Linear Equations

Abstract:
By the end of this section, you will be able to:

* Recognize the relation between the solutions of an equation and its graph
* Graph a linear equation by plotting points
* Graph vertical and horizontal lines

Recognize the Relation Between the Solutions of an Equation and its Graph

In Use the Rectangular Coordinate System, we found a few solutions to the equation $3x+2y=6$. They are listed in the table below. So, the ordered pairs $\left(0,3\right)$, $\left(2,0\right)$, $\left(1,\frac{3}{2}\right)$, $\left(4,-3\right)$, are some solutions to the equation$3x+2y=6$. We can plot these solutions in the rectangular coordinate system as shown on the graph at right.

Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The method we used at the start of this section to graph is called plotting points, or the Point-Plotting Method.

Graph Vertical and Horizontal Lines

Can we graph an equation with only one variable? Just $x$ and no $y,$ or just $y$ without an $x?$ How will we make a table of values to get the points to plot?

Graphing with Intercepts

Abstract:
By the end of this section, you will be able to:

* Identify the intercepts on a graph
* Find the intercepts from an equation of a line
* Graph a line using the intercepts
* Choose the most convenient method to graph a line

Identify the Intercepts on a Graph

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points.

Find the Intercepts from an Equation of a Line

Recognizing that the $x\text{-intercept}$ occurs when $y$ is zero and that the $y\text{-intercept}$ occurs when $x$ is zero gives us a method to find the intercepts of a line from its equation. To find the $x\text{-intercept,}$ let $y=0$ and solve for $x.$ To find the $y\text{-intercept},$ let $x=0$ and solve for $y.$

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line.

Choose the Most Convenient Method to Graph a Line

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them.

Understand Slope of a Line

Use Geoboards to Model Slope

In this section, we will explore the concepts of slope.

Find the Slope of a Line from its Graph

Now we’ll look at some graphs on a coordinate grid to find their slopes. The method will be very similar to what we just modeled on our geoboards.

Find the Slope of Horizontal and Vertical Lines

Do you remember what was special about horizontal and vertical lines? Their equations had just one variable.

Use the Slope Formula to find the Slope of a Line between Two Points

Sometimes we need to find the slope of a line between two points and we might not 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 there is a way to find the slope without graphing.

Graph a Line Given a Point and the Slope

In this chapter, we graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

Solve Slope Applications

At the beginning of this section, we said there are many applications of slope in the real world. Let’s look at a few now.

Cumulative Review

Powers and Roots Tables

Geometric Formulas