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Unformatted text preview: 1 Elementary Algebra - MTH 0661 Collection edited by: Sharon Bailey and Robert Connolly Content authors: OpenStax Based on: Elementary Algebra < ;. Online: < ; This selection and arrangement of content as a collection is copyrighted by Sharon Bailey and Robert Connolly. Creative Commons Attribution License 4.0 Collection structure revised: 2018/07/25 PDF Generated: 2019/02/13 17:13:47 For copyright and attribution information for the modules contained in this collection, see the "Attributions" section at the end of the collection. 2 This OpenStax book is available for free at Table of Contents Preface 1 Solving Linear Equations and Inequalities 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2 4 Use the Rectangular Coordinate System 211 Graph Linear Equations in Two Variables 232 Graph with Intercepts 252 Understand Slope of a Line 267 Use the Slope–Intercept Form of an Equation of a Line Find the Equation of a Line 321 Graphs of Linear Inequalities 339 443 481 Add and Subtract Polynomials 481 Use Multiplication Properties of Exponents 495 Multiply Polynomials 509 Special Products 525 Divide Monomials 538 Divide Polynomials 556 Integer Exponents and Scientific Notation 568 Factoring 6.1 6.2 6.3 6.4 6.5 6.6 295 373 Solve Systems of Equations by Graphing 373 Solve Systems of Equations by Substitution 394 Solve Systems of Equations by Elimination 410 Solve Applications with Systems of Equations 425 Solve Mixture Applications with Systems of Equations Graphing Systems of Linear Inequalities 456 Polynomials 5.1 5.2 5.3 5.4 5.5 5.6 5.7 6 211 Systems of Linear Equations 4.1 4.2 4.3 4.4 4.5 4.6 5 103 Use a Problem-Solving Strategy 103 Solve Percent Applications 120 Solve Mixture Applications 138 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem Solve Uniform Motion Applications 177 Solve Applications with Linear Inequalities 190 Graphs 3.1 3.2 3.3 3.4 3.5 3.6 3.7 5 Solve Equations Using the Subtraction and Addition Properties of Equality 5 Solve Equations using the Division and Multiplication Properties of Equality 20 Solve Equations with Variables and Constants on Both Sides 34 Use a General Strategy to Solve Linear Equations 44 Solve Equations with Fractions or Decimals 57 Solve a Formula for a Specific Variable 68 Solve Linear Inequalities 78 Math Models 2.1 2.2 2.3 2.4 2.5 2.6 3 1 599 Greatest Common Factor and Factor by Grouping 599 Factor Quadratic Trinomials with Leading Coefficient 1 613 Factor Quadratic Trinomials with Leading Coefficient Other than 1 Factor Special Products 644 General Strategy for Factoring Polynomials 660 Quadratic Equations 671 Index 749 626 154 This OpenStax book is available for free at Preface 1 PREFACE Welcome to Elementary Algebra, an OpenStax resource. This textbook was written to increase student access to highquality 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. Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on openstax.org. 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, 2 Preface 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 PDF and the Web View version of the text, 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. This OpenStax book is available for free at Preface 3 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, 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. 4 Preface 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 federallyfunded 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 This OpenStax book is available for free at Chapter 1 Solving Linear Equations and Inequalities 5 SOLVING LINEAR EQUATIONS AND INEQUALITIES 1 Figure 1.1 The rocks in this formation must remain perfectly balanced around the center for the formation to hold its shape. Chapter Outline 1.1 Solve Equations Using the Subtraction and Addition Properties of Equality 1.2 Solve Equations using the Division and Multiplication Properties of Equality 1.3 Solve Equations with Variables and Constants on Both Sides 1.4 Use a General Strategy to Solve Linear Equations 1.5 Solve Equations with Fractions or Decimals 1.6 Solve a Formula for a Specific Variable 1.7 Solve Linear Inequalities Introduction If we carefully placed more rocks of equal weight on both sides of this formation, it would still balance. Similarly, the expressions in an equation remain balanced when we add the same quantity to both sides of the equation. In this chapter, we will solve equations, remembering that what we do to one side of the equation, we must also do to the other side. 1.1 Solve Equations Using the Subtraction and Addition Properties of Equality Learning Objectives 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 Be Prepared! Before you get started, take this readiness quiz. 1. Evaluate x + 4 when x = −3 . If you missed this problem, review m60212 ( ) . 2. Evaluate 15 − y when y = −5 . If you missed this problem, review m60212 ( ) . 6 Chapter 1 Solving Linear Equations and Inequalities 3. Simplify 4(4n + 1) − 15n . If you missed this problem, review m60218 ( ) . 4. Translate into algebra “5 is less than x .” If you missed this problem, review m60247 ( ) . 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! Solution of an equation A solution of an equation is a value of a variable that makes a true statement when substituted into the equation. HOW TO : : TO DETERMINE WHETHER A NUMBER IS A SOLUTION TO AN EQUATION. Step 1. Substitute the number in for the variable in the equation. Step 2. Simplify the expressions on both sides of the equation. Step 3. Determine whether the resulting equation is true (the left side is equal to the right side) ◦ If it is true, the number is a solution. ◦ If it is not true, the number is not a solution. EXAMPLE 1.1 Determine whether x = 3 is a solution of 4x − 2 = 2x + 1 . 2 Solution Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable. Multiply. Subtract. Since x = 3 results in a true equation (4 is in fact equal to 4), 3 is a solution to the equation 4x − 2 = 2x + 1 . 2 2 TRY IT : : 1.1 TRY IT : : 1.2 Is y = 4 a solution of 9y + 2 = 6y + 3 ? 3 Is y = 7 a solution of 5y + 3 = 10...
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