Unformatted text preview: 1 Elementary Algebra - MTH 0661
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1 Solving Linear Equations and Inequalities
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
6.6 295 373 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
5.7 6 211 Systems of Linear Equations
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
3.7 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
2.6 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
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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
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
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
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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
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
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
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 .
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.
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 .
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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