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Intermediate Algebra 2e SENIOR CONTRIBUTING AUTHORS LYNN MARECEK, SANTA ANA COLLEGE ANDREA HONEYCUTT MATHIS, NORTHEAST MISSISSIPPI COMMUNITY COLLEGE OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit https://openstax.org Individual print copies and bulk orders can be purchased through our website ©2020 Rice University Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0) Under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows: - - - - If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you must retain on every page the following attribution: “Access for free at openstax.org.” If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: “Access for free at openstax.org.” If you 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future of education openstax.org Table of Contents Preface Foundations 1.1 1.2 1.3 1.4 1.5 227 Graph Linear Equations in Two Variables 227 Slope of a Line 256 Find the Equation of a Line 281 Graph Linear Inequalities in Two Variables 298 Relations and Functions 317 Graphs of Functions 332 371 Solve Systems of Linear Equations with Two Variables 371 Solve Applications with Systems of Equations 393 Solve Mixture Applications with Systems of Equations 411 Solve Systems of Equations with Three Variables 425 Solve Systems of Equations Using Matrices 438 Solve Systems of Equations Using Determinants 452 Graphing Systems of Linear Inequalities 466 Add and Subtract Polynomials 493 Properties of Exponents and Scientific Notation Multiply Polynomials 532 Dividing Polynomials 549 Factoring 6.1 6.2 6.3 6.4 6.5 Use a General Strategy to Solve Linear Equations 99 Use a Problem Solving Strategy 116 Solve a Formula for a Specific Variable 135 Solve Mixture and Uniform Motion Applications 152 Solve Linear Inequalities 171 Solve Compound Inequalities 190 Solve Absolute Value Inequalities 201 Polynomials and Polynomial Functions 5.1 5.2 5.3 5.4 99 Systems of Linear Equations 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Use the Language of Algebra Integers 24 Fractions 43 Decimals 57 Properties of Real Numbers 74 Graphs and Functions 3.1 3.2 3.3 3.4 3.5 3.6 Solving Linear Equations 2.1 2.2 2.3 2.4 2.5 2.6 2.7 507 575 Greatest Common Factor and Factor by Grouping Factor Trinomials 584 Factor Special Products 603 General Strategy for Factoring Polynomials 617 Polynomial Equations 627 Rational Expressions and Functions 7.1 7.2 7.3 7.4 7.5 7.6 493 653 Multiply and Divide Rational Expressions 653 Add and Subtract Rational Expressions 670 Simplify Complex Rational Expressions 685 Solve Rational Equations 697 Solve Applications with Rational Equations 712 Solve Rational Inequalities 735 575 Roots and Radicals 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 Simplify Expressions with Roots 757 Simplify Radical Expressions 773 Simplify Rational Exponents 792 Add, Subtract, and Multiply Radical Expressions Divide Radical Expressions 821 Solve Radical Equations 834 Use Radicals in Functions 848 Use the Complex Number System 857 Quadratic Equations and Functions 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 10 883 Exponential and Logarithmic Functions 1015 Finding Composite and Inverse Functions 1015 Evaluate and Graph Exponential Functions 1033 Evaluate and Graph Logarithmic Functions 1049 Use the Properties of Logarithms 1066 Solve Exponential and Logarithmic Equations 1079 Conics 11.1 11.2 11.3 11.4 11.5 12 808 Solve Quadratic Equations Using the Square Root Property 883 Solve Quadratic Equations by Completing the Square 896 Solve Quadratic Equations Using the Quadratic Formula 911 Solve Quadratic Equations in Quadratic Form 925 Solve Applications of Quadratic Equations 933 Graph Quadratic Functions Using Properties 947 Graph Quadratic Functions Using Transformations 973 Solve Quadratic Inequalities 994 10.1 10.2 10.3 10.4 10.5 11 757 1103 Distance and Midpoint Formulas; Circles 1103 Parabolas 1118 Ellipses 1139 Hyperbolas 1156 Solve Systems of Nonlinear Equations 1170 Sequences, Series and Binomial Theorem 12.1 12.2 12.3 12.4 Sequences 1197 Arithmetic Sequences 1210 Geometric Sequences and Series Binomial Theorem 1238 Index 1221 1385 This OpenStax book is available for free at http://cnx.org/content/col31234/1.3 1197 Preface PREFACE Welcome to Intermediate Algebra 2e, 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 35 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 Intermediate Algebra 2e 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 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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 Intermediate Algebra 2e Intermediate Algebra 2e is designed to meet the scope and sequence requirements of a one-semester Intermediate 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 Intermediate Algebra 2e continues the philosophies and pedagogical features of Prealgebra and Elementary Algebra, by Lynn Marecek and MaryAnne Anthony-Smith By introducing the concepts and vocabulary of algebra in a nurturing, nonthreatening environment while also addressing the needs of students with diverse backgrounds and learning styles, the book helps students gain confidence in their ability to succeed in the course and become successful college students The material is presented as a sequence of small, and clear steps to conceptual understanding The order of topics was carefully planned to emphasize the logical progression throughout the course and to facilitate a thorough understanding of each concept As new ideas are presented, they are explicitly related to previous topics Chapter 1: Foundations Chapter reviews arithmetic operations with whole numbers, integers, fractions, decimals and real numbers, 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 solve linear equations using the Properties of Equality and a general strategy They use a problem-solving strategy to solve number, percent, mixture and uniform motion applications Solving Preface a formula for a specific variable, and also solving both linear and compound inequalities is presented Chapter 3: Graphs and Functions Chapter covers the rectangular coordinate system where students learn to plot graph linear equations in two variables, graph with intercepts, 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 The chapter also introduces relations and functions as well as graphing of functions Chapter 4: Systems of Linear Equations Chapter 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 Systems of equations are also solved using matrices and determinants Chapter 5: Polynomials and Polynomial Functions In Chapter 5, 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 6: Factoring In Chapter 6, students learn the process of factoring expressions and see how factoring is used to solve quadratic equations Chapter 7: Rational Expressions and Functions In Chapter 7, students work with rational expressions, solve rational equations and use them to solve problems in a variety of applications, and solve rational inequalities Chapter 8: Roots and Radical In Chapter 8, students simplify radical expressions, rational exponents, perform operations on radical expressions, and solve radical equations Radical functions and the complex number system are introduced Chapter 9: Quadratic Equations In Chapter 9, students use various methods to solve quadratic equations and equations in quadratic form and learn how to use them in applications Students will graph quadratic functions using their properties and by transformations Chapter 10: Exponential and Logarithmic Functions In Chapter 10, students find composite and inverse functions, evaluate, graph, and solve both exponential and logarithmic functions Chapter 11: Conics In Chapter 11, the properties and graphs of circles, parabolas, ellipses and hyperbolas are presented Students also solve applications using the conics and solve systems of nonlinear equations Chapter 12: Sequences, Series and the Binomial Theorem In Chapter 12, students are introduced to sequences, arithmetic sequences, geometric sequences and series and the binomial theorem All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents Changes to the Second Edition The Intermediate Algebra 2e revision focused on mathematical clarity and accuracy Every Example, Try-It, Section Exercise, Review Exercise, and Practice Test item was reviewed by multiple faculty experts, and then verified by authors This intensive effort resulted in hundreds of changes to the text, problem language, answers, instructor solutions, and graphics However, OpenStax and our authors are aware of the difficulties posed by shifting problem and exercise numbers when textbooks are revised In an effort to make the transition to the 2nd edition as seamless as possible, we have minimized any shifting of exercise numbers For example, instead of deleting or adding problems where necessary, we replaced problems in order to keep the numbering intact As a result, in nearly all chapters, there will be no shifting of exercise numbers; in the chapters where shifting does occur, it will be minor Faculty and course coordinators should be able to use the new edition in a straightforward manner Also, to increase convenience, answers to the Be Prepared Exercises will now appear in the regular solutions manuals, rather than as a separate resource A detailed transition guide is available as an instructor resource at openstax.org Pedagogical Foundation and Features Examples Each learning objective is supported by one or more worked examples, which 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 This OpenStax book is available for free at http://cnx.org/content/col31234/1.3 ... Coverage and Scope Intermediate Algebra 2e continues the philosophies and pedagogical features of Prealgebra and Elementary Algebra, by Lynn Marecek and MaryAnne Anthony-Smith By introducing the... for a low cost in print About Intermediate Algebra 2e Intermediate Algebra 2e is designed to meet the scope and sequence requirements of a one-semester Intermediate Algebra course The book’s organization... Intermediate Algebra 2e SENIOR CONTRIBUTING AUTHORS LYNN MARECEK, SANTA ANA COLLEGE ANDREA HONEYCUTT MATHIS, NORTHEAST MISSISSIPPI COMMUNITY COLLEGE