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Elementary Algebra

SENIOR CONTRIBUTING AUTHORS LYNN MARECEK, SANTA ANA COLLEGE MARYANNE ANTHONY-SMITH, FORMERLY OF SANTA ANA COLLEGE

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Table of Contents

Preface 1

Foundations 5 1.1 Introduction to Whole Numbers 5 1.2 Use the Language of Algebra 21 1.3 Add and Subtract Integers 40 1.4 Multiply and Divide Integers 61 1.5 Visualize Fractions 76 1.6 Add and Subtract Fractions 92 1.7 Decimals 107 1.8 The Real Numbers 126 1.9 Properties of Real Numbers 142 1.10 Systems of Measurement 160

Solving Linear Equations and Inequalities 197 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality 197 2.2 Solve Equations using the Division and Multiplication Properties of Equality 212 2.3 Solve Equations with Variables and Constants on Both Sides 226 2.4 Use a General Strategy to Solve Linear Equations 236 2.5 Solve Equations with Fractions or Decimals 249 2.6 Solve a Formula for a Specific Variable 260 2.7 Solve Linear Inequalities 270

Math Models 295 3.1 Use a Problem-Solving Strategy 295 3.2 Solve Percent Applications 312 3.3 Solve Mixture Applications 330 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem 346 3.5 Solve Uniform Motion Applications 369 3.6 Solve Applications with Linear Inequalities 382

Graphs 403 4.1 Use the Rectangular Coordinate System 403 4.2 Graph Linear Equations in Two Variables 424 4.3 Graph with Intercepts 444 4.4 Understand Slope of a Line 459 4.5 Use the Slope–Intercept Form of an Equation of a Line 486 4.6 Find the Equation of a Line 512 4.7 Graphs of Linear Inequalities 530

Systems of Linear Equations 565 5.1 Solve Systems of Equations by Graphing 565 5.2 Solve Systems of Equations by Substitution 586 5.3 Solve Systems of Equations by Elimination 602 5.4 Solve Applications with Systems of Equations 617 5.5 Solve Mixture Applications with Systems of Equations 635 5.6 Graphing Systems of Linear Inequalities 648

Polynomials 673 6.1 Add and Subtract Polynomials 673 6.2 Use Multiplication Properties of Exponents 687 6.3 Multiply Polynomials 701 6.4 Special Products 717 6.5 Divide Monomials 730 6.6 Divide Polynomials 748 6.7 Integer Exponents and Scientific Notation 760

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Factoring 789 7.1 Greatest Common Factor and Factor by Grouping 789 7.2 Factor Quadratic Trinomials with Leading Coefficient 1 803 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1 816 7.4 Factor Special Products 834 7.5 General Strategy for Factoring Polynomials 850 7.6 Quadratic Equations 861

Rational Expressions and Equations 883 8.1 Simplify Rational Expressions 883 8.2 Multiply and Divide Rational Expressions 901 8.3 Add and Subtract Rational Expressions with a Common Denominator 914 8.4 Add and Subtract Rational Expressions with Unlike Denominators 923 8.5 Simplify Complex Rational Expressions 937 8.6 Solve Rational Equations 950 8.7 Solve Proportion and Similar Figure Applications 965 8.8 Solve Uniform Motion and Work Applications 981 8.9 Use Direct and Inverse Variation 991

Roots and Radicals 1013 9.1 Simplify and Use Square Roots 1013 9.2 Simplify Square Roots 1023 9.3 Add and Subtract Square Roots 1036 9.4 Multiply Square Roots 1046 9.5 Divide Square Roots 1060 9.6 Solve Equations with Square Roots 1074 9.7 Higher Roots 1091 9.8 Rational Exponents 1107

Quadratic Equations 1137 10.1 Solve Quadratic Equations Using the Square Root Property 1137 10.2 Solve Quadratic Equations by Completing the Square 1149 10.3 Solve Quadratic Equations Using the Quadratic Formula 1165 10.4 Solve Applications Modeled by Quadratic Equations 1179 10.5 Graphing Quadratic Equations 1190

Index 1309

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PREFACE

Welcome to Elementary Algebra, an OpenStax resource. This textbook was written to increase student access to high- quality learning materials, maintaining highest standards of academic rigor at little to no cost.

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 25 books for college and AP courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learning outcomes through personalized educational paths, 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 Customization Elementary Algebra is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book. Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on openstax.org for more information.

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Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.

About Elementary Algebra Elementary Algebra 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.

Coverage and Scope Elementary Algebra follows a nontraditional approach in its presentation of content. Building on the content in Prealgebra, the material 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 through the course and to facilitate a thorough understanding of each concept. As new ideas are presented, they are explicitly related to previous topics.

Chapter 1: Foundations Chapter 1 reviews arithmetic operations with whole numbers, integers, fractions, and decimals, to give the student a solid base that will support their study of algebra. Chapter 2: Solving Linear Equations and Inequalities In Chapter 2, students learn to verify a solution of an equation, solve equations using the Subtraction and Addition Properties of Equality, solve equations using the Multiplication and Division Properties of Equality, solve equations with variables and constants on both sides, use a general strategy to solve linear equations, solve equations with fractions or decimals, solve a formula for a specific variable, and solve linear inequalities. Chapter 3: Math Models Once students have learned the skills needed to solve equations, they apply these skills in Chapter 3 to solve word and number problems. Chapter 4: Graphs Chapter 4 covers the rectangular coordinate system, which is the basis for most consumer graphs. Students learn to plot points on a rectangular coordinate system, graph linear equations in two variables, graph with intercepts,

Preface 1

understand slope of a line, use the slope-intercept form of an equation of a line, find the equation of a line, and create graphs of linear inequalities. Chapter 5: Systems of Linear Equations Chapter 5 covers solving systems of equations by graphing, substitution, and elimination; solving applications with systems of equations, solving mixture applications with systems of equations, and graphing systems of linear inequalities. Chapter 6: Polynomials In Chapter 6, students learn how to add and subtract polynomials, use multiplication properties of exponents, multiply polynomials, use special products, divide monomials and polynomials, and understand integer exponents and scientific notation. Chapter 7: Factoring In Chapter 7, students explore the process of factoring expressions and see how factoring is used to solve certain types of equations. Chapter 8: Rational Expressions and Equations In Chapter 8, students work with rational expressions, solve rational equations, and use them to solve problems in a variety of applications. Chapter 9: Roots and Radical In Chapter 9, students are introduced to and learn to apply the properties of square roots, and extend these concepts to higher order roots and rational exponents. Chapter 10: Quadratic Equations In Chapter 10, students study the properties of quadratic equations, solve and graph them. They also learn how to apply them as models of various situations.

All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.

Key Features and Boxes Examples Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the solution, spelling out the steps along the way. Finally (for select Examples), we show students how to check the solution. Most Examples are written in a two-column format, with explanation on the left and math on the right to mimic the way that instructors “talk through” examples as they write on the board in class. Be Prepared! Each section, beginning with Section 2.1, starts with a few “Be Prepared!” exercises so that students can determine if they have mastered the prerequisite skills for the section. Reference is made to specific Examples from previous sections so students who need further review can easily find explanations. Answers to these exercises can be found in the supplemental resources that accompany this title. Try It

The Try It feature includes a pair of exercises that immediately follow an Example, providing the student with an immediate opportunity to solve a similar problem. In the Web View version of the text, students can click an Answer link directly below the question to check their understanding. In the PDF, answers to the Try It exercises are located in the Answer Key. How To

How To feature typically follows the Try It exercises and outlines the series of steps for how to solve the problem in the preceding Example. Media

The Media icon appears at the conclusion of each section, just prior to the Self Check. This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section. Disclaimer: While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany Elementary Algebra. Self Check The Self Check includes the learning objectives for the section so that students can self-assess their mastery and make concrete plans to improve.

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Art Program Elementary Algebra contains many figures and illustrations. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions.

Section Exercises and Chapter Review Section Exercises Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. Exercise sets are named Practice Makes Perfect to encourage completion of homework assignments.

Exercises correlate to the learning objectives. This facilitates assignment of personalized study plans based on individual student needs. Exercises are carefully sequenced to promote building of skills. Values for constants and coefficients were chosen to practice and reinforce arithmetic facts. Even and odd-numbered exercises are paired. Exercises parallel and extend the text examples and use the same instructions as the examples to help students easily recognize the connection. Applications are drawn from many everyday experiences, as well as those traditionally found in college math texts. Everyday Math highlights practical situations using the concepts from that particular section Writing Exercises are included in every exercise set to encourage conceptual understanding, critical thinking, and literacy.

Chapter Review Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.

Key Terms provide a formal definition for each bold-faced term in the chapter. Key Concepts summarize the most important ideas introduced in each section, linking back to the relevant Example(s) in case students need to review. Chapter Review Exercises include practice problems that recall the most important concepts from each section. Practice Test includes additional problems assessing the most important learning objectives from the chapter. Answer Key includes the answers to all Try It exercises and every other exercise from the Section Exercises, Chapter Review Exercises, and Practice Test.

Additional Resources Student and Instructor Resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, manipulative mathematics worksheets, Links to Literacy assignments, and an answer key to Be Prepared Exercises. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book.

Partner Resources OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on openstax.org.

About the Authors Senior Contributing Authors Lynn Marecek and MaryAnne Anthony-Smith have been teaching mathematics at Santa Ana College for many years and have worked together on several projects aimed at improving student learning in developmental math courses. They are the authors of Strategies for Success: Study Skills for the College Math Student.

Preface 3

Lynn Marecek, Santa Ana College Lynn Marecek has focused her career on meeting the needs of developmental math students. At Santa Ana College, she has been awarded the Distinguished Faculty Award, Innovation Award, and the Curriculum Development Award four times. She is a Coordinator of Freshman Experience Program, the Department Facilitator for Redesign, and a member of the Student Success and Equity Committee, and the Basic Skills Initiative Task Force. Lynn holds a bachelor’s degree from Valparaiso University and master’s degrees from Purdue University and National University. MaryAnne Anthony-Smith, Santa Ana College MaryAnne Anthony-Smith was a mathematics professor at Santa Ana College for 39 years, until her retirement in June, 2015. She has been awarded the Distinguished Faculty Award, as well as the Professional Development, Curriculum Development, and Professional Achievement awards. MaryAnne has served as department chair, acting dean, chair of the professional development committee, institutional researcher, and faculty coordinator on several state and federally- funded grants. She is the community college coordinator of California’s Mathematics Diagnostic Testing Project, a member of AMATYC’s Placement and Assessment Committee. She earned her bachelor’s degree from the University of California San Diego and master’s degrees from San Diego State and Pepperdine Universities.

Reviewers Jay Abramson, Arizona State University Bryan Blount, Kentucky Wesleyan College Gale Burtch, Ivy Tech Community College Tamara Carter, Texas A&M University Danny Clarke, Truckee Meadows Community College Michael Cohen, Hofstra University Christina Cornejo, Erie Community College Denise Cutler, Bay de Noc Community College Lance Hemlow, Raritan Valley Community College John Kalliongis, Saint Louis Iniversity Stephanie Krehl, Mid-South Community College Laurie Lindstrom, Bay de Noc Community College Beverly Mackie, Lone Star College System Allen Miller, Northeast Lakeview College Christian Roldán-Johnson, College of Lake County Community College Martha Sandoval-Martinez, Santa Ana College Gowribalan Vamadeva, University of Cincinnati Blue Ash College Kim Watts, North Lake College Libby Watts, Tidewater Community College Allen Wolmer, Atlantic Jewish Academy John Zarske, Santa Ana College

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Figure 1.1 In order to be structurally sound, the foundation of a building must be carefully constructed.

Chapter Outline 1.1 Introduction to Whole Numbers 1.2 Use the Language of Algebra 1.3 Add and Subtract Integers 1.4 Multiply and Divide Integers 1.5 Visualize Fractions 1.6 Add and Subtract Fractions 1.7 Decimals 1.8 The Real Numbers 1.9 Properties of Real Numbers 1.10 Systems of Measurement

Introduction Just like a building needs a firm foundation to support it, your study of algebra needs to have a firm foundation. To ensure this, we begin this book with a review of arithmetic operations with whole numbers, integers, fractions, and decimals, so that you have a solid base that will support your study of algebra.

1.1 Introduction to Whole Numbers Learning Objectives By the end of this section, you will be able to:

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

Be Prepared!

A more thorough introduction to the topics covered in this section can be found in Prealgebra in the chapters Whole Numbers and The Language of Algebra.

As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

FOUNDATIONS1

Chapter 1 Foundations 5

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

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly. We can visualize counting numbers and whole numbers on a number line (see Figure 1.2).

Figure 1.2 The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. While this number line shows only the whole numbers 0 through 6, the numbers keep going without end.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Number Line-Part 1” will help you develop a better understanding of the counting numbers and the whole numbers.

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 1.3 shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Figure 1.3 The number 5,278,194 is shown in the chart. The digit 5 is in the millions place. The digit 2 is in the hundred-thousands place. The digit 7 is in the ten-thousands place. The digit 8 is in the thousands place. The digit 1 is in the hundreds place. The digit 9 is in the tens place. The digit 4 is in the ones place.

EXAMPLE 1.1

In the number 63,407,218, find the place value of each digit:

ⓐ 7 ⓑ 0 ⓒ 1 ⓓ 6 ⓔ 3 Solution

Place the number in the place value chart:

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ⓐ The 7 is in the thousands place. ⓑ The 0 is in the ten thousands place. ⓒ The 1 is in the tens place. ⓓ The 6 is in the ten-millions place. ⓔ The 3 is in the millions place.

TRY IT : : 1.1 For the number 27,493,615, find the place value of each digit:

ⓐ 2 ⓑ 1 ⓒ 4 ⓓ 7 ⓔ 5

TRY IT : : 1.2 For the number 519,711,641,328, find the place value of each digit:

ⓐ 9 ⓑ 4 ⓒ 2 ⓓ 6 ⓔ 7

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 1.4). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Figure 1.4

EXAMPLE 1.2

Name the number 8,165,432,098,710 using words.

Solution Name the number in each period, followed by the period name.

HOW TO : : NAME A WHOLE NUMBER IN WORDS.

Start at the left and name the number in each period, followed by the period name. Put commas in the number to separate the periods. Do not name the ones period.

Step 1. Step 2. Step 3.

Chapter 1 Foundations 7

Put the commas in to separate the periods. So, 8, 165, 432, 098, 710 is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

TRY IT : : 1.3 Name the number 9, 258, 137, 904, 061 using words.

TRY IT : : 1.4 Name the number 17, 864, 325, 619, 004 using words.

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers,