Preface

About OpenStax

OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 35 books for college and AP courses used by hundreds of thousands of students. OpenStax Tutor, our low-cost personalized learning tool, is being piloted in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

About OpenStax Resources

About *Elementary Algebra*

*Elementary Algebra 2e* is designed to meet the scope and sequence requirements of a one-semester elementary algebra course. The book’s organization makes it easy to adapt to a variety of course syllabi. The text expands on 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.

Additional Resources

About the Authors

Foundations

Introduction

Introduction to Whole Numbers

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

* Use place value with whole numbers
* Identify multiples and apply divisibility tests
* Find prime factorizations and least common multiples

Use Place Value with Whole Numbers

The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting number**s**. Counting numbers are also called *natural numbers*. If we add zero to the counting numbers, we get the set of whole number**s**.

Identify Multiples and Apply Divisibility Tests

The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of 2 and a counting number.

Find Prime Factorizations and Least Common Multiples

In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if *m* is a multiple of *n*, we can say that *m* is divisible by *n*. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way.

Use the Language of Algebra

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

* Use variables and algebraic symbols
* Simplify expressions using the order of operations
* Evaluate an expression
* Identify and combine like terms
* Translate an English phrase to an algebraic expression

Use Variables and Algebraic Symbols

Suppose this year Greg is 20 years old and Alex is 23. You know that Alex is 3 years older than Greg. When Greg was 12, Alex was 15. When Greg is 35, Alex will be 38. No matter what Greg’s age is, Alex’s age will always be 3 years more, right? In the language of algebra, we say that Greg’s age and Alex’s age are variable**s** and the 3 is a constant. The ages change (“vary”) but the 3 years between them always stays the same (“constant”). Since Greg’s age and Alex’s age will always differ by 3 years, 3 is the *constant*.

Simplify Expressions Using the Order of Operations

To simplify an 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:

Evaluate an Expression

In the last few examples, we simplified expressions using the order of operations. Now we’ll evaluate some expressions—again following the order of operations. To **evaluate an expression** means to find the value of the expression when the variable is replaced by a given number.

Identify and Combine Like Terms

Algebraic expressions are made up of terms. A **term** is a constant, or the product of a constant and one or more variables.

Translate an English Phrase to an Algebraic Expression

In the last section, we listed many operation symbols that are used in algebra, then we translated expressions and equations into English phrases and sentences. Now we’ll reverse the process. We’ll translate English phrases into algebraic expressions. The symbols and variables we’ve talked about will help us do that. [link] summarizes them.

Add and Subtract Integers

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

* Use negatives and opposites
* Simplify: expressions with absolute value
* Add integers
* Subtract integers

Use Negatives and Opposites

Our work so far has only included the counting numbers and the whole numbers. But if you have ever experienced a temperature below zero or accidentally overdrawn your checking account, you are already familiar with negative numbers. **Negative numbers** are numbers less than $0.$ The negative numbers are to the left of zero on the number line. See [link].

Simplify: Expressions with Absolute Value

We saw that numbers such as $2{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}-2$ are opposites because they are the same distance from 0 on the number line. They are both two units from 0. The distance between 0 and any number on the number line is called the **absolute value** of that number.

Add Integers

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

Subtract Integers

We will continue to use counters to model the subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers.

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 English phrases to algebraic expressions
* Use integers in applications

Multiply Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we will use the model just to help us discover the pattern.

Divide Integers

What about division? Division is the inverse operation of multiplication. So, $15÷3=5$ because $5·3=15.$ In words, this expression says that 15 can be divided into three groups of five each because adding five three times gives 15. Look at some examples of multiplying integers, to figure out the rules for dividing integers.

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember My Dear Aunt Sally?

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

Visualize Fractions

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

* Find equivalent fractions
* Simplify fractions
* Multiply fractions
* Divide fractions
* Simplify expressions written with a fraction bar
* Translate phrases to expressions with fractions

Find Equivalent Fractions

**Fractions** are a way to represent parts of a whole. The fraction $\frac{1}{3}$ means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See [link]. The fraction $\frac{2}{3}$ represents two of three equal parts. In the fraction $\frac{2}{3},$ the 2 is called the numerator and the 3 is called the denominator.

Simplify Fractions

A fraction is considered **simplified** if there are no common factors, other than 1, in its numerator and denominator.

Multiply Fractions

Many people find multiplying and dividing fractions easier than adding and subtracting fractions. So we will start with fraction multiplication.

Divide Fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, we need some vocabulary.

Simplify Expressions with a Fraction Bar

The line that separates the numerator from the denominator in a fraction is called a fraction bar. A fraction bar acts as grouping symbol. The order of operations then tells us to simplify the numerator and then the denominator. Then we divide.

Translate Phrases to Expressions with Fractions

Now that we have done some work with fractions, we are ready to translate phrases that would result in expressions with fractions.

Add and Subtract Fractions

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

* Add or subtract fractions with a common denominator
* Add or subtract fractions with different denominators
* Use the order of operations to simplify complex fractions
* Evaluate variable expressions with fractions

Add or Subtract Fractions with a Common Denominator

When we multiplied fractions, we just multiplied the numerators and multiplied the denominators right straight across. To add or subtract fractions, they must have a common denominator.

Add or Subtract Fractions with Different Denominators

As we have seen, to add or subtract fractions, their denominators must be the same. The least common denominator (LCD) of two fractions is the smallest number that can be used as a common denominator of the fractions. The LCD of the two fractions is the least common multiple (LCM) of their denominators.

Use the Order of Operations to Simplify Complex Fractions

We have seen that a complex fraction is a fraction in which the numerator or denominator contains a fraction. The fraction bar indicates division. We simplified the complex fraction $\frac{\frac{3}{4}}{\frac{5}{8}}$ by dividing $\frac{3}{4}$ by $\frac{5}{8}.$

Evaluate Variable Expressions with Fractions

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

Decimals

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

* Name and write decimals
* Round decimals
* Add and subtract decimals
* Multiply and divide decimals
* Convert decimals, fractions, and percents

Name and Write Decimals

**Decimals** are another way of writing fractions whose denominators are powers of 10.

Round Decimals

Rounding decimals is very much like rounding whole numbers. We will round decimals with a method based on the one we used to round whole numbers.

Add and Subtract Decimals

To add or subtract decimals, we line up the decimal points. By lining up the decimal points this way, we can add or subtract the corresponding place values. We then add or subtract the numbers as if they were whole numbers and then place the decimal point in the sum.

Multiply and Divide Decimals

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

Convert Decimals, Fractions, and Percents

We convert decimals into fractions by identifying the place value of the last (farthest right) digit. In the decimal 0.03 the 3 is in the hundredths place, so 100 is the denominator of the fraction equivalent to 0.03.

The Real Numbers

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

* Simplify expressions with square roots
* Identify integers, rational numbers, irrational numbers, and real numbers
* Locate fractions on the number line
* Locate decimals on the number line

Simplify Expressions with Square Roots

Remember that when a number *n* is multiplied by itself, we write ${n}^{2}$ and read it “n squared.” The result is called the square of *n*. For example,

Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers

We have already described numbers as *counting number**s*, *whole number**s*, and *integers*. What is the difference between these types of numbers?

Locate Fractions on the Number Line

The last time we looked at the number line, it only had positive and negative integers on it. We now want to include fractions and decimals on it.

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.

Properties of Real Numbers

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

* Use the commutative and associative properties
* Use the identity and inverse properties of addition and multiplication
* Use the properties of zero
* Simplify expressions using the distributive property

Use the Commutative and Associative Properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

Use the Identity and Inverse Properties of Addition and Multiplication

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

Use the Properties of Zero

The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need \$9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

Systems of Measurement

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

* Make unit conversions in the US system
* Use mixed units of measurement in the US system
* Make unit conversions in the metric system
* Use mixed units of measurement in the metric system
* Convert between the US 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 U.S. 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

We often use mixed units of measurement in everyday situations. Suppose Joe is 5 feet 10 inches tall, stays at work for 7 hours and 45 minutes, and then eats a 1 pound 2 ounce steak for dinner—all these measurements have mixed units.

Make Unit Conversions in the Metric System

In the metric system, units are related by powers of 10. The roots words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is 1,000 meters; the prefix *kilo* means *thousand*. One centimeter is $\frac{1}{100}$ of a meter, 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 US system. But it may be easier because of the relation of the units to the powers of 10. Make sure to add or subtract like units.

Convert Between the U.S. and the Metric Systems of Measurement

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

Convert between Fahrenheit and Celsius Temperatures

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

Solving Linear Equations and Inequalities

Introduction

Solve Equations Using the Subtraction and Addition Properties of Equality

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

* Verify a solution of an equation
* Solve equations using the Subtraction and Addition Properties of Equality
* Solve equations that require simplification
* Translate to an equation and solve
* Translate and solve applications

Verify a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in [link]. Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?

Solve Equations That Require Simplification

In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.

Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign. [link] shows us some of the words that are commonly used.

Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

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 require simplification
* Translate to an equation and solve
* Translate and solve applications

Solve Equations Using the Division and Multiplication Properties of Equality

You may have noticed that all of the equations we have solved so far have been of the form $x+a=b$ or $x-a=b$. We were able to isolate the variable by adding or subtracting the constant term on the side of the equation with the variable. Now we will see how to solve equations that have a variable multiplied by a constant and so will require division to isolate the variable.

Many equations start out more complicated than the ones we have been working with.

Translate to an Equation and Solve

In the next few examples, we will translate sentences into equations and then solve the equations. You might want to review the translation table in the previous chapter.

Translate and Solve Applications

To solve applications using the Division and Multiplication Properties of Equality, we will follow the same steps we used in the last section. We will restate the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve.

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 with Constants on Both Sides

In all the equations we have solved so far, all the variable terms were on only one side of the equation with the constants on the other side. This does not happen all the time—so now we will learn to solve equations in which the variable terms, or constant terms, or both are on both sides of the equation.

Solve Equations with Variables on Both Sides

What if there are variables on both sides of the equation? For equations like this, begin as we did above—choose a “variable” side and a “constant” side, and then use the subtraction and addition properties of equality to collect all variables on one side and all constants on the other side.

The next example will be the first to have variables and constants on both sides of the equation. It may take several steps to solve this equation, so we need a clear and organized strategy.

Use a General Strategy to Solve Linear Equations

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

* Solve equations using a general strategy
* Classify equations

Solve Equations Using the General Strategy

Until now we have dealt with solving one specific form of a linear equation. It is time now to lay out one overall strategy that can be used to solve any linear equation. Some equations we solve will not require all these steps to solve, but many will.

Classify Equations

Consider the equation we solved at the start of the last section, $7x+8=-13$. The solution we found was $x=-3$. This means the equation $7x+8=-13$ is true when we replace the variable, *x*, with the value $-3$. We showed this when we checked the solution $x=-3$ and evaluated $7x+8=-13$ for $x=-3$.

Solve Equations with Fractions or Decimals

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, $\frac{1}{8}x+\frac{1}{2}=\frac{1}{4}$.

Solve Equations with Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, $0.3=\frac{3}{10}$ and $0.17=\frac{17}{100}$. So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

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

Solve a Formula for a Specific Variable

You are probably familiar with some geometry formulas. A formula is a mathematical description of the relationship between variables. Formulas are also used in the sciences, such as chemistry, physics, and biology. In medicine they are used for calculations for dispensing medicine or determining body mass index. Spreadsheet programs rely on formulas to make calculations. It is important to be familiar with formulas and be able to manipulate them easily.

Solve Linear Inequalities

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

* Graph inequalities on the number line
* Solve inequalities using the Subtraction and Addition Properties of inequality
* Solve inequalities using the Division and Multiplication Properties of inequality
* Solve inequalities that require simplification
* Translate to an inequality and solve

Graph Inequalities on the Number Line

Do you remember what it means for a number to be a solution to an equation? A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

Solve Inequalities using the Subtraction and Addition Properties of Inequality

The Subtraction and Addition Properties of Equality state that if two quantities are equal, when we add or subtract the same amount from both quantities, the results will be equal.

Solve Inequalities using the Division and Multiplication Properties of Inequality

The Division and Multiplication Properties of Equality state that if two quantities are equal, when we divide or multiply both quantities by the same amount, the results will also be equal (provided we don’t divide by 0).

Solve Inequalities That Require Simplification

Most inequalities will take more than one step to solve. We follow the same steps we used in the general strategy for solving linear equations, but be sure to pay close attention during multiplication or division.

Translate to an Inequality and Solve

To translate English sentences into inequalities, we need to recognize the phrases that indicate the inequality. Some words are easy, like ‘more than’ and ‘less than’. But others are not as obvious.

Math Models

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

“If you think you can… or think you can’t… you’re right.”—Henry Ford

Use a Problem-Solving Strategy for Word Problems

We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.

Solve Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Solve Percent Applications

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

* Translate and solve basic percent equations
* Solve percent applications
* Find percent increase and percent decrease
* Solve simple interest applications
* Solve applications with discount or mark-up

Translate and Solve Basic Percent Equations

We will solve percent equations using the methods we used to solve equations with fractions or decimals. Without the tools of algebra, the best method available to solve percent problems was by setting them up as proportions. Now as an algebra student, you can just translate English sentences into algebraic equations and then solve the equations.

Solve Applications of Percent

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

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 Simple Interest Applications

Do you know that banks pay you to keep your money? The money a customer puts in the bank is called the principal, *P*, and the money the bank pays the customer is called the interest. The interest is computed as a certain percent of the principal; called the rate of interest, *r*. We usually express rate of interest as a percent per year, and we calculate it by using the decimal equivalent of the percent. The variable *t*, (for *time*) represents the number of years the money is in the account.

Solve Applications with Discount or Mark-up

Applications of discount are very common in retail settings. When you buy an item on sale, the original price has been discounted by some dollar amount. The discount rate, usually given as a percent, is used to determine the amount of the discount. To determine the amount of discount, we multiply the discount rate by the original price.

Solve Mixture Applications

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

* Solve coin word problems
* Solve ticket and stamp word problems
* Solve mixture word problems
* Use the mixture model to solve investment problems using simple interest

Solve Coin Word Problems

In mixture problems, we will have two or more items with different values to combine together. The mixture model is used by grocers and bartenders to make sure they set fair prices for the products they sell. Many other professionals, like chemists, investment bankers, and landscapers also use the mixture model.

Solve Ticket and Stamp Word Problems

Problems involving tickets or stamps are very much like coin problems. Each type of ticket and stamp has a value, just like each type of coin does. So to solve these problems, we will follow the same steps we used to solve coin problems.

Solve Mixture Word Problems

Now we’ll solve some more general applications of the mixture model. Grocers and bartenders use the mixture model to set a fair price for a product made from mixing two or more ingredients. Financial planners use the mixture model when they invest money in a variety of accounts and want to find the overall interest rate. Landscape designers use the mixture model when they have an assortment of plants and a fixed budget, and event coordinators do the same when choosing appetizers and entrees for a banquet.

Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem

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

* Solve applications using properties of triangles
* Use the Pythagorean Theorem
* Solve applications using rectangle properties

Solve Applications Using Properties of Triangles

In this section we will use some common geometry formulas. We will adapt our problem-solving strategy so that we can solve geometry applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve shapes of some sort, most people find it helpful to draw a figure and label it with the given information. We will include this in the first step of the problem solving strategy for geometry applications.

Use the Pythagorean Theorem

We have learned how the measures of the angles of a triangle relate to each other. Now, we will learn how the lengths of the sides relate to each other. An important property that describes the relationship among the lengths of the three sides of a right triangle is called the Pythagorean Theorem. This theorem has been used around the world since ancient times. It is named after the Greek philosopher and mathematician, Pythagoras, who lived around 500 BC.

Solve Applications Using Rectangle Properties

You may already be familiar with the properties of rectangles. Rectangles have four sides and four right $\left(90\text{°}\right)$ angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, *L*, and its adjacent side as the width, *W*.

Solve Uniform Motion Applications

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

* Solve uniform motion applications

Solve Uniform Motion Applications

When planning a road trip, it often helps to know how long it will take to reach the destination or how far to travel each day. We would use the distance, rate, and time formula, $D=rt,$ which we have already seen.

Solve Applications with Linear Inequalities

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

* Solve applications with linear inequalities

Many real-life situations require us to solve inequalities. In fact, inequality applications are so common that we often do not even realize we are doing algebra. For example, how many gallons of gas can be put in the car for \$20? Is the rent on an apartment affordable? Is there enough time before class to go get lunch, eat it, and return? How much money should each family member’s holiday gift cost without going over budget?

Graphs

Introduction

Use the Rectangular Coordinate System

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

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

Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the *xy*-plane or the ‘coordinate plane’.

Verify Solutions to an Equation in Two Variables

Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. The process of solving an equation ended with a statement like $x=4$. (Then, you checked the solution by substituting back into the equation.)

Complete a Table of Solutions to a Linear Equation in Two Variables

In the examples above, we substituted the *x*- and *y*-values of a given ordered pair to determine whether or not it was a solution to a linear equation. But how do you find the ordered pairs if they are not given? It’s easier than you might think—you can just pick a value for $x$ and then solve the equation for $y$. Or, pick a value for $y$ and then solve for $x$.

Find Solutions to a Linear Equation

To find a solution to a linear equation, you really can pick *any* number you want to substitute into the equation for $x$ or $y.$ But since you’ll need to use that number to solve for the other variable it’s a good idea to choose a number that’s easy to work with.

Graph  Linear Equations in Two Variables

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

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

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

In the previous section, we found several solutions to the equation $3x+2y=6$. They are listed in [link]. So, the ordered pairs $\left(0,3\right)$, $\left(2,0\right)$, and $\left(1,\frac{3}{2}\right)$ are some solutions to the equation $3x+2y=6$. We can plot these solutions in the rectangular coordinate system as shown in [link].

Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The method we used to graph $3x+2y=6$ 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?

Graph with Intercepts

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

* Identify the x - and y - intercepts on a graph
* Find the x - and y - intercepts from an equation of a line
* Graph a line using the intercepts

Identify the *x*- and *y*- Intercepts on a Graph

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.

Find the *x*- and *y*- Intercepts from an Equation of a Line

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

Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the *x*- and *y*- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.

Understand Slope of a Line

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

* Use geoboards to model slope
* Use $m=\frac{rise}{run}$ to find the slope of a line from its graph
* Find the slope of horizontal and vertical lines
* Use the slope formula to find the slope of a line between two points
* Graph a line given a point and the slope
* Solve slope applications

Use Geoboards to Model Slope

A geoboard is a board with a grid of pegs on it. Using rubber bands on a geoboard gives us a concrete way to model lines on a coordinate grid. By stretching a rubber band between two pegs on a geoboard, we can discover how to find the slope of a line.

Use $m=\frac{\text{rise}}{\text{run}}$ to Find the Slope of a Line from its Graph

Now, we’ll look at some graphs on the $xy$-coordinate plane and see how 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’ll need to find the slope of a line between two points when we don’t have a graph to count out the rise and the run. We could plot the points on grid paper, then count out the rise and the run, but as we’ll see, there is a way to find the slope without graphing. Before we get to it, we need to introduce some algebraic notation.

Graph a Line Given a Point and the Slope

Up to now, in this chapter, we have graphed lines by plotting points, by using intercepts, and by recognizing horizontal and vertical lines.

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.

Use the Slope-Intercept Form of an Equation of a Line

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

* Recognize the relation between the graph and the slope–intercept form of an equation of a line
* Identify the slope and y-intercept form of an equation of a line
* Graph a line using its slope and intercept
* Choose the most convenient method to graph a line
* Graph and interpret applications of slope–intercept
* Use slopes to identify parallel lines
* Use slopes to identify perpendicular lines

Recognize the Relation Between the Graph and the Slope–Intercept Form of an Equation of a Line

We have graphed linear equations by plotting points, using intercepts, recognizing horizontal and vertical lines, and using the point–slope method. Once we see how an equation in slope–intercept form and its graph are related, we’ll have one more method we can use to graph lines.

Identify the Slope and *y*-Intercept From an Equation of a Line

In Understand Slope of a Line, we graphed a line using the slope and a point. When we are given an equation in slope–intercept form, we can use the *y*-intercept as the point, and then count out the slope from there. Let’s practice finding the values of the slope and *y*-intercept from the equation of a line.

Graph a Line Using its Slope and Intercept

Now that we know how to find the slope and *y*-intercept of a line from its equation, we can graph the line by plotting the *y*-intercept and then using the slope to find another point.

Choose the Most Convenient Method to Graph a Line

Now that we have seen several methods we can use to graph lines, how do we know which method to use for a given equation?

Graph and Interpret Applications of Slope–Intercept

Many real-world applications are modeled by linear equations. We will take a look at a few applications here so you can see how equations written in slope–intercept form relate to real-world situations.

Use Slopes to Identify Parallel Lines

The slope of a line indicates how steep the line is and whether it rises or falls as we read it from left to right. Two lines that have the same slope are called parallel lines. Parallel lines never intersect.

Use Slopes to Identify Perpendicular Lines

Let’s look at the lines whose equations are $y=\frac{1}{4}x-1$ and $y=-4x+2$, shown in [link].

Find the Equation of a Line

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

* Find an equation of the line given the slope and $y$-intercept
* Find an equation of the line given the slope and a point
* Find an equation of the line given two points
* Find an equation of a line parallel to a given line
* Find an equation of a line perpendicular to a given line

Find an Equation of the Line Given the Slope and *y*-Intercept

We can easily determine the slope and intercept of a line if the equation was written in slope–intercept form, $y=mx+b.$ Now, we will do the reverse—we will start with the slope and *y*-intercept and use them to find the equation of the line.

Find an Equation of the Line Given the Slope and a Point

Finding an equation of a line using the slope–intercept form of the equation works well when you are given the slope and *y*-intercept or when you read them off a graph. But what happens when you have another point instead of the *y*-intercept?

Find an Equation of the Line Given Two Points

When real-world data is collected, a linear model can be created from two data points. In the next example we’ll see how to find an equation of a line when just two points are given.

Find an Equation of a Line Parallel to a Given Line

Suppose we need to find an equation of a line that passes through a specific point and is parallel to a given line. We can use the fact that parallel lines have the same slope. So we will have a point and the slope—just what we need to use the point–slope equation.

Find an Equation of a Line Perpendicular to a Given Line

Now, let’s consider perpendicular lines. Suppose we need to find a line passing through a specific point and which is perpendicular to a given line. We can use the fact that perpendicular lines have slopes that are negative reciprocals. We will again use the point–slope equation, like we did with parallel lines.

Graphs of Linear Inequalities

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

* Verify solutions to an inequality in two variables
* Recognize the relation between the solutions of an inequality and its graph
* Graph linear inequalities

Verify Solutions to an Inequality in Two Variables

We have learned how to solve inequalities in one variable. Now, we will look at inequalities in two variables. Inequalities in two variables have many applications. If you ran a business, for example, you would want your revenue to be greater than your costs—so that your business would make a profit.

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

Now, we will look at how the solutions of an inequality relate to its graph.

Graph Linear Inequalities

Now, we’re ready to put all this together to graph linear inequalities.

Systems of Linear Equations

Introduction

Solve Systems of Equations by Graphing

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

* Determine whether an ordered pair is a solution of a system of equations
* Solve a system of linear equations by graphing
* Determine the number of solutions of linear system
* Solve applications of systems of equations by graphing

Determine Whether an Ordered Pair is a Solution of a System of Equations

In Solving Linear Equations and Inequalities we learned how to solve linear equations with one variable. Remember that the solution of an equation is a value of the variable that makes a true statement when substituted into the equation.

Solve a System of Linear Equations by Graphing

In this chapter we will use three methods to solve a system of linear equations. The first method we’ll use is graphing.

Determine the Number of Solutions of a Linear System

There will be times when we will want to know how many solutions there will be to a system of linear equations, but we might not actually have to find the solution. It will be helpful to determine this without graphing.

Solve Applications of Systems of Equations by Graphing

We will use the same problem solving strategy we used in Math Models to set up and solve applications of systems of linear equations. We’ll modify the strategy slightly here to make it appropriate for systems of equations.

Solving Systems of Equations by Substitution

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

* Solve a system of equations by substitution
* Solve applications of systems of equations by substitution

Solve a System of Equations by Substitution

We will use the same system we used first for graphing.

Solve Applications of Systems of Equations by Substitution

We’ll copy here the problem solving strategy we used in the Solving Systems of Equations by Graphing section for solving systems of equations. Now that we know how to solve systems by substitution, that’s what we’ll do in Step 5.

Solve Systems of Equations by Elimination

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

* Solve a system of equations by elimination
* Solve applications of systems of equations by elimination
* Choose the most convenient method to solve a system of linear equations

Solve a System of Equations by Elimination

The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.

Solve Applications of Systems of Equations by Elimination

Some applications problems translate directly into equations in standard form, so we will use the elimination method to solve them. As before, we use our Problem Solving Strategy to help us stay focused and organized.

Choose the Most Convenient Method to Solve a System of Linear Equations

When you will have to solve a system of linear equations in a later math class, you will usually not be told which method to use. You will need to make that decision yourself. So you’ll want to choose the method that is easiest to do and minimizes your chance of making mistakes.

Solve Applications with Systems of Equations

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

* Translate to a system of equations
* Solve direct translation applications
* Solve geometry applications
* Solve uniform motion applications

Translate to a System of Equations

Many of the problems we solved in earlier applications related two quantities. Here are two of the examples from the chapter on Math Models.

Solve Direct Translation Applications

We set up, but did not solve, the systems of equations in [link] and [link] Now we’ll translate a situation to a system of equations and then solve it.

Solve Geometry Applications

When we learned about Math Models, we solved geometry applications using properties of triangles and rectangles. Now we’ll add to our list some properties of angles.

Solve Uniform Motion Applications

We used a table to organize the information in uniform motion problems when we introduced them earlier. We’ll continue using the table here. The basic equation was *D* = *rt* where *D* is the distance travelled, *r* is the rate, and *t* is the time.

Solve Mixture Applications with Systems of Equations

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

* Solve mixture applications
* Solve interest applications

Solve Mixture Applications

When we solved mixture applications with coins and tickets earlier, we started by creating a table so we could organize the information. For a coin example with nickels and dimes, the table looked like this:

Solve Interest Applications

The formula to model interest applications is *I* = *Prt*. Interest, *I*, is the product of the principal, *P*, the rate, *r*, and the time, *t*. In our work here, we will calculate the interest earned in one year, so *t* will be 1.

Graphing Systems of Linear Inequalities

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

* Determine whether an ordered pair is a solution of a system of linear inequalities
* Solve a system of linear inequalities by graphing
* Solve applications of systems of inequalities

Determine Whether an Ordered Pair is a Solution of a System of Linear Inequalities

The definition of a system of linear inequalities is very similar to the definition of a system of linear equations.

Solve a System of Linear Inequalities by Graphing

The solution to a single linear inequality is the region on one side of the boundary line that contains all the points that make the inequality true. The solution to a system of two linear inequalities is a region that contains the solutions to both inequalities. To find this region, we will graph each inequality separately and then locate the region where they are both true. The solution is always shown as a graph.

Solve Applications of Systems of Inequalities

The first thing we’ll need to do to solve applications of systems of inequalities is to translate each condition into an inequality. Then we graph the system as we did above to see the region that contains the solutions. Many situations will be realistic only if both variables are positive, so their graphs will only show Quadrant I.

Polynomials

Introduction

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

You have learned that a *term* is a constant or the product of a constant and one or more variables. When it is of the form $a{x}^{m}$, where $a$ is a constant and $m$ is a whole number, it is called a monomial. Some examples of monomial are $8,-2{x}^{2},4{y}^{3},{\rule{0.2em}{0ex}}\text{and}{\rule{0.2em}{0ex}}11{z}^{7}$.

Determine the Degree of Polynomials

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

Add and Subtract Monomials

You have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficient.

Add and Subtract Polynomials

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Evaluate a Polynomial for a Given Value

We have already learned how to evaluate expressions. Since polynomials are expressions, we’ll follow the same procedures to evaluate a polynomial. We will substitute the given value for the variable and then simplify using the order of operations.

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 for Exponents
* Simplify expressions using the Power Property for 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 2 by itself 4 times, so ${2}^{4}$ means $2·2·2·2$.

Simplify Expressions Using the Product Property for Exponents

You have seen that when you combine like terms by adding and subtracting, you need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

Simplify Expressions Using the Power Property for Exponents

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

Simplify Expressions Using the Product to a Power Property

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

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 of exponents to multiply 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

We have used the Distributive Property to simplify expressions like $2\left(x-3\right)$. 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.

Multiply a Binomial by a Binomial

Just like there are different ways to represent multiplication of numbers, there are several methods that can be used to multiply a binomial times a binomial. We will start by using the Distributive Property.

Multiply a Trinomial by a Binomial

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

Special Products

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

* Square a binomial using the Binomial Squares Pattern
* Multiply conjugates using the Product of Conjugates Pattern
* Recognize and use the appropriate special product pattern

Square a Binomial Using the Binomial Squares Pattern

Mathematicians like to look for patterns that will make their work easier. A good example of this is squaring binomials. While you can always get the product by writing the binomial twice and using the methods of the last section, there is less work to do if you learn to use a pattern.

Multiply Conjugates Using the Product of Conjugates Pattern

We just saw a pattern for squaring binomials that we can use to make multiplying some binomials easier. Similarly, there is a pattern for another product of binomials. But before we get to it, we need to introduce some vocabulary.

Recognize and Use the Appropriate Special Product Pattern

We just developed special product patterns for Binomial Squares and for the Product of Conjugates. The products look similar, so it is important to recognize when it is appropriate to use each of these patterns and to notice how they differ. Look at the two patterns together and note their similarities and differences.

Divide Monomials

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

* Simplify expressions using the Quotient Property for 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 for Exponents

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

Simplify Expressions with an Exponent of Zero

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

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Divide Polynomials

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

* Divide a polynomial by a monomial
* Divide a polynomial by a binomial

Divide a Polynomial by a Monomial

In the last section, you learned how to divide a monomial by a monomial. As you continue to build up your knowledge of polynomials the next procedure is to divide a polynomial of two or more terms by a monomial.

Divide a Polynomial by a Binomial

To divide a polynomial by a binomial, we follow a procedure very similar to long division of numbers. So let’s look carefully the steps we take when we divide a 3-digit number, 875, by a 2-digit number, 25.

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

We saw that the Quotient Property for Exponents introduced earlier in this chapter, has two forms depending on whether the exponent is larger in the numerator or the denominator.

Simplify Expressions with Integer Exponents

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

Convert from Decimal Notation to Scientific Notation

Remember working with place value for whole numbers and decimals? Our number system is based on powers of 10. We use tens, hundreds, thousands, and so on. Our decimal numbers are also based on powers of tens—tenths, hundredths, thousandths, and so on. Consider the numbers 4,000 and $0.004$. We know that 4,000 means $4{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}1,000$ and 0.004 means $4{\rule{0.2em}{0ex}}×{\rule{0.2em}{0ex}}\frac{1}{1,000}$.

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

Astronomers use very large numbers to describe distances in the universe and ages of stars and planets. Chemists use very small numbers to describe the size of an atom or the charge on an electron. When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Factoring

Introduction

Greatest Common Factor and Factor by Grouping

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
* Factor by grouping

Find the Greatest Common Factor of Two or More Expressions

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

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 $2·6{\rule{0.2em}{0ex}}\text{or}{\rule{0.2em}{0ex}}3·4\right),$ in algebra, it can be useful to represent a polynomial in factored form. One way to do this is by finding the GCF of all the terms. Remember, we multiply a polynomial by a monomial as follows:

Factor by Grouping

When there is no common factor of all the terms of a polynomial, look for a common factor in just some of the terms. When there are four terms, a good way to start is by separating the polynomial into two parts with two terms in each part. Then look for the GCF in each part. If the polynomial can be factored, you will find a common factor emerges from both parts.

Factor Trinomials of the Form x2+bx+c

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

* Factor trinomials of the form ${x}^{2}+bx+c$
* Factor trinomials of the form ${x}^{2}+bxy+c{y}^{2}$

Factor Trinomials of the Form *x*2 + *bx* + *c*

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.

Factor Trinomials of the Form *x*2 + *bxy* + *cy*2

Sometimes you’ll need to factor trinomials of the form ${x}^{2}+bxy+c{y}^{2}$ with two variables, such as ${x}^{2}+12xy+36{y}^{2}.$ The first term, ${x}^{2}$, is the product of the first terms of the binomial factors, $x·x$. The ${y}^{2}$ in the last term means that the second terms of the binomial factors must each contain *y*. To get the coefficients *b* and *c*, you use the same process summarized in the previous objective.

Factor Trinomials of the Form ax2+bx+c

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

* Recognize a preliminary strategy to factor polynomials completely
* Factor trinomials of the form $a{x}^{2}+bx+c$ with a GCF
* Factor trinomials using trial and error
* Factor trinomials using the ‘ac’ method

Recognize a Preliminary Strategy for Factoring

Let’s summarize where we are so far with factoring polynomials. In the first two sections of this chapter, we used three methods of factoring: factoring the GCF, factoring by grouping, and factoring a trinomial by “undoing” FOIL. More methods will follow as you continue in this chapter, as well as later in your studies of algebra.

Factor Trinomials of the form *ax*2 + *bx* + *c* with a GCF

Now that we have organized what we’ve covered so far, we are ready to factor trinomials whose leading coefficient is not 1, trinomials of the form $a{x}^{2}+bx+c$.

Factor Trinomials using Trial and Error

What happens when the leading coefficient is not 1 and there is no GCF? There are several methods that can be used to factor these trinomials. First we will use the Trial and Error method.

Factor Trinomials using the “ac” Method

Another way to factor trinomials of the form $a{x}^{2}+bx+c$ is the “ac” method. (The “ac” method is sometimes called the grouping method.) The “ac” method is actually an extension of the methods you used in the last section to factor trinomials with leading coefficient one. This method is very structured (that is step-by-step), and it always works!

Factor Special Products

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

* Factor perfect square trinomials
* Factor differences of squares
* Factor sums and differences of cubes
* Choose method to factor a polynomial completely

Factor Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. You can square a binomial by using FOIL, but using the Binomial Squares pattern you saw in a previous chapter saves you a step. Let’s review the Binomial Squares pattern by squaring a binomial using FOIL.

Factor Differences of Squares

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

Factor Sums and Differences of Cubes

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.

General Strategy for Factoring Polynomials

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

* Recognize and use the appropriate method to factor a polynomial completely

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

You have now become acquainted with all the methods of factoring that you will need in this course. (In your next algebra course, more methods will be added to your repertoire.) The figure below summarizes all the factoring methods we have covered. [link] outlines a strategy you should use when factoring polynomials.

Quadratic Equations

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

* Solve quadratic equations by using the Zero Product Property
* Solve quadratic equations factoring
* Solve applications modeled by quadratic equations

Solve Quadratic Equations Using the Zero Product Property

We will first solve some quadratic equations by using the Zero Product Property. The Zero Product Property says that if the product of two quantities is zero, it must be that at least one of the quantities is zero. The only way to get a product equal to zero is to multiply by zero itself.

Solve Quadratic Equations by Factoring

Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we must be sure to start with the quadratic equation in standard form, $a{x}^{2}+bx+c=0$. Then we can factor the expression on the left.

Solve Applications Modeled by Quadratic Equations

The problem solving strategy we used earlier for applications that translate to linear equations will work just as well for applications that translate to quadratic equations. We will copy the problem solving strategy here so we can use it for reference.

Rational Expressions and Equations

Introduction

Simplify Rational Expressions

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

* Determine the values for which a rational expression is undefined
* Evaluate rational expressions
* Simplify rational expressions
* Simplify rational expressions with opposite factors

Determine the Values for Which a Rational Expression is Undefined

When we work with a numerical fraction, it is easy to avoid dividing by zero, because we can see the number in the denominator. In order to avoid dividing by zero in a rational expression, we must not allow values of the variable that will make the denominator be zero.

Evaluate Rational Expressions

To evaluate a rational expression, we substitute values of the variables into the expression and simplify, just as we have for many other expressions in this book.

Simplify Rational Expressions

Just like a fraction is considered simplified if there are no common factors, other than 1, in its numerator and denominator, a rational expression is *simplified* if it has no common factors, other than 1, in its numerator and denominator.

Simplify Rational Expressions with Opposite Factors

Now we will see how to simplify a rational expression whose numerator and denominator have opposite factors. Let’s start with a numerical fraction, say $\frac{7}{-7}$. We know this fraction simplifies to $-1$. We also recognize that the numerator and denominator are opposites.

Multiply and Divide Rational Expressions

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

* Multiply rational expressions
* Divide rational expressions

Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Add and Subtract Rational Expressions with a Common Denominator

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

* Add rational expressions with a common denominator
* Subtract rational expressions with a common denominator
* Add and subtract rational expressions whose denominators are opposites

Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by $\frac{-1}{-1}$.

Add and Subtract Rational Expressions with Unlike Denominators

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

* Find the least common denominator of rational expressions
* Find equivalent rational expressions
* Add rational expressions with different denominators
* Subtract rational expressions with different denominators

Find the Least Common Denominator of Rational Expressions

When we add or subtract rational expressions with unlike denominators we will need to get common denominators. If we review the procedure we used with numerical fractions, we will know what to do with rational expressions.

Find Equivalent Rational Expressions

When we add numerical fractions, once we find the LCD, we rewrite each fraction as an equivalent fraction with the LCD.

Add Rational Expressions with Different Denominators

Now we have all the steps we need to add rational expressions with different denominators. As we have done previously, we will do one example of adding numerical fractions first.

Subtract Rational Expressions with Different Denominators

The process we use to subtract rational expressions with different denominators is the same as for addition. We just have to be very careful of the signs when subtracting the numerators.

Simplify Complex Rational Expressions

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

* Simplify a complex rational expression by writing it as division
* Simplify a complex rational expression by using the LCD

Simplify a Complex Rational Expression by Writing it as Division

We have already seen this complex rational expression earlier in this chapter.

Simplify a Complex Rational Expression by Using the LCD

We “cleared” the fractions by multiplying by the LCD when we solved equations with fractions. We can use that strategy here to simplify complex rational expressions. We will multiply the numerator and denominator by LCD of all the rational expressions.

Solve Rational Equations

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

* Solve rational equations
* Solve a rational equation for a specific variable

Solve Rational Equations

We have already solved linear equations that contained fractions. We found the LCD of all the fractions in the equation and then multiplied both sides of the equation by the LCD to “clear” the fractions.

Solve a Rational Equation for a Specific Variable

When we solved linear equations, we learned how to solve a formula for a specific variable. Many formulas used in business, science, economics, and other fields use rational equations to model the relation between two or more variables. We will now see how to solve a rational equation for a specific variable.

Solve Proportion and Similar Figure Applications

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

* Solve proportions
* Solve similar figure applications

Solve Proportions

When two rational expressions are equal, the equation relating them is called a *proportion*.

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be *similar.* One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

Solve Uniform Motion and Work Applications

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

* Solve uniform motion applications
* Solve work applications

Solve Uniform Motion Applications

We have solved uniform motion problems using the formula $D=rt$ in previous chapters. We used a table like the one below to organize the information and lead us to the equation.

Solve Work Applications

Suppose Pete can paint a room in 10 hours. If he works at a steady pace, in 1 hour he would paint $\frac{1}{10}$ of the room. If Alicia would take 8 hours to paint the same room, then in 1 hour she would paint $\frac{1}{8}$ of the room. How long would it take Pete and Alicia to paint the room if they worked together (and didn’t interfere with each other’s progress)?

Use Direct and Inverse Variation

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

* Solve direct variation problems
* Solve inverse variation problems

Solve Direct Variation Problems

Lindsay gets paid \$15 per hour at her job. If we let *s* be her salary and *h* be the number of hours she has worked, we could model this situation with the equation

Solve Inverse Variation Problems

Many applications involve two variable that *vary inversely*. As one variable increases, the other decreases. The equation that relates them is $y=\frac{k}{x}$.

Roots and Radicals

Introduction

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

Simplify Expressions with Square Roots

Remember that when a number $n$ is multiplied by itself, we write ${n}^{2}$ and read it “n squared.” For example, ${15}^{2}$ reads as “15 squared,” and 225 is called the square of 15, since ${15}^{2}=225$.

Estimate Square Roots

So far we have only considered square roots of perfect square numbers. The square roots of other numbers are not whole numbers. Look at [link] below.

Approximate Square Roots

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find them. Find the $\sqrt{x}$ key on your calculator. You will use this key to approximate square roots.

Simplify Variable Expressions with Square Roots

What if we have to find a square root of an expression with a variable? Consider $\sqrt{9{x}^{2}}$. Can you think of an expression whose square is $9{x}^{2}$?

Simplify Square Roots

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

* Use the Product Property to simplify square roots
* Use the Quotient Property to simplify square roots

Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that ${\left(ab\right)}^{m}={a}^{m}{b}^{m}$. The corresponding property of square roots says that $\sqrt{ab}=\sqrt{a}·\sqrt{b}$.

Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A *perfect square fraction* is a fraction in which both the numerator and the denominator are perfect squares.

Add and Subtract Square Roots

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

* Add and subtract like square roots
* Add and subtract square roots that need simplification

Add and Subtract Like Square Roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

Multiply Square Roots

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

* Multiply square roots
* Use polynomial multiplication to multiply square roots

Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

Divide Square Roots

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

* Divide square roots
* Rationalize a one-term denominator
* Rationalize a two-term denominator

Divide Square Roots

We know that we simplify fractions by removing factors common to the numerator and the denominator. When we have a fraction with a square root in the numerator, we first simplify the square root. Then we can look for common factors.

Rationalize a One Term Denominator

Before the calculator became a tool of everyday life, tables of square roots were used to find approximate values of square roots. [link] shows a portion of a table of squares and square roots. Square roots are approximated to five decimal places in this table.

Rationalize a Two-Term Denominator

When the denominator of a fraction is a sum or difference with square roots, we use the Product of Conjugates pattern to rationalize the denominator.

Solve Equations with Square Roots

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

* Solve radical equations
* Use square roots in applications

Solve Radical Equations

In this section we will solve equations that have the variable in the radicand of a square root. Equations of this type are called radical equations.

Use Square Roots in Applications

As you progress through your college courses, you’ll encounter formulas that include square roots in many disciplines. We have already used formulas to solve geometry applications.

Higher Roots

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

* Simplify expressions with higher roots
* Use the Product Property to simplify expressions with higher roots
* Use the Quotient Property to simplify expressions with higher roots
* Add and subtract higher roots

Simplify Expressions with Higher Roots

Up to now, in this chapter we have worked with squares and square roots. We will now extend our work to include higher powers and higher roots.

Use the Product Property to Simplify Expressions with Higher Roots

We will simplify expressions with higher roots in much the same way as we simplified expressions with square roots. An *n*th root is considered simplified if it has no factors of ${m}^{n}$.

Use the Quotient Property to Simplify Expressions with Higher Roots

We can simplify higher roots with quotients in the same way we simplified square roots. First we simplify any fractions inside the radical.

Add and Subtract Higher Roots

We can add and subtract higher roots like we added and subtracted square roots. First we provide a formal definition of like radicals.

Rational Exponents

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

* Simplify expressions with ${a}^{\frac{1}{n}}$
* Simplify expressions with ${a}^{\frac{m}{n}}$
* Use the Laws of Exponents to simply expressions with rational exponents

Simplify Expressions with ${a}^{\frac{1}{n}}$

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

Simplify Expressions with ${a}^{\frac{m}{n}}$

Let’s work with the Power Property for Exponents some more.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

Quadratic Equations

Introduction

Solve Quadratic Equations Using the Square Root Property

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

* Solve quadratic equations of the form $a{x}^{2}=k$ using the Square Root Property
* Solve quadratic equations of the form $a{\left(x-h\right)}^{2}=k$ using the Square Root Property

Solve Quadratic Equations of the Form *ax*2 = *k* Using the Square Root Property

We have already solved some quadratic equations by factoring. Let’s review how we used factoring to solve the quadratic equation ${x}^{2}=9$.

Solve Quadratic Equations of the Form *a*(*x* − *h*)2 = *k* Using the Square Root Property

We can use the Square Root Property to solve an equation like ${\left(x-3\right)}^{2}=16$, too. We will treat the whole binomial, $\left(x-3\right)$, as the quadratic term.

Solve Quadratic Equations by Completing the Square

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

* Complete the square of a binomial expression
* Solve quadratic equations of the form ${x}^{2}+bx+c=0$ by completing the square
* Solve quadratic equations of the form $a{x}^{2}+bx+c=0$ by completing the square

Complete The Square of a Binomial Expression

In the last section, we were able to use the Square Root Property to solve the equation ${\left(y-7\right)}^{2}=12$ because the left side was a perfect square.

Solve Quadratic Equations of the Form *x*2 + *bx* + *c* = 0 by completing the square

In solving equations, we must always do the same thing to both sides of the equation. This is true, of course, when we solve a quadratic equation by completing the square, too. When we add a term to one side of the equation to make a perfect square trinomial, we must also add the same term to the other side of the equation.

Solve Quadratic Equations of the form *ax*2 + *bx* + *c* = 0 by completing the square

The process of completing the square works best when the leading coefficient is one, so the left side of the equation is of the form ${x}^{2}+bx+c$. If the ${x}^{2}$ term has a coefficient, we take some preliminary steps to make the coefficient equal to one.

Solve Quadratic Equations Using the Quadratic Formula

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

* Solve quadratic equations using the quadratic formula
* Use the discriminant to predict the number of solutions of a quadratic equation
* Identify the most appropriate method to use to solve a quadratic equation

Use the Discriminant to Predict the Number of Solutions of a Quadratic Equation

When we solved the quadratic equations in the previous examples, sometimes we got two solutions, sometimes one solution, sometimes no real solutions. Is there a way to predict the number of solutions to a quadratic equation without actually solving the equation?

Identify the Most Appropriate Method to Use to Solve a Quadratic Equation

We have used four methods to solve quadratic equations:

Solve Applications Modeled by Quadratic Equations

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

* Solve applications modeled by Quadratic Equations

Solve Applications of the Quadratic Formula

We solved some applications that are modeled by quadratic equations earlier, when the only method we had to solve them was factoring. Now that we have more methods to solve quadratic equations, we will take another look at applications. To get us started, we will copy our usual Problem Solving Strategy here so we can follow the steps.

Graphing Quadratic Equations in Two Variables

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

* Recognize the graph of a quadratic equation in two variables
* Find the axis of symmetry and vertex of a parabola
* Find the intercepts of a parabola
* Graph quadratic equations in two variables
* Solve maximum and minimum applications

Recognize the Graph of a Quadratic Equation in Two Variables

We have graphed equations of the form $Ax+By=C$. We called equations like this linear equations because their graphs are straight lines.

Find the Axis of Symmetry and Vertex of a Parabola

Look again at [link]. Do you see that we could fold each parabola in half and that one side would lie on top of the other? The ‘fold line’ is a line of symmetry. We call it the axis of symmetry of the parabola.

Find the Intercepts of a Parabola

When we graphed linear equations, we often used the *x*- and *y*-intercepts to help us graph the lines. Finding the coordinates of the intercepts will help us to graph parabolas, too.

Graph Quadratic Equations in Two Variables

Now, we have all the pieces we need in order to graph a quadratic equation in two variables. We just need to put them together. In the next example, we will see how to do this.

Solve Maximum and Minimum Applications

Knowing that the vertex of a parabola is the lowest or highest point of the parabola gives us an easy way to determine the minimum or maximum value of a quadratic equation. The *y*-coordinate of the vertex is the minimum *y*-value of a parabola that opens upward. It is the maximum *y*-value of a parabola that opens downward. See [link].