COVER
TITLE PAGE
COPYRIGHT
ACKNOWLEDGMENTS
INTRODUCTION
1 BASIC CONCEPTS
1. 1.1 Addition, Subtraction, Multiplication, Division
2. 1.2 Order of Operations
3. 1.3 Sets and Properties of Numbers
4. CHAPTER 1 TEST
5. CHAPTER 1 SOLUTIONS
6. CHAPTER 1 TEST SOLUTIONS
2 FRACTIONS
1. 2.1 Basic Operations
2. 2.2 Simplifying Fractions
3. CHAPTER 2 TEST
4. CHAPTER 2 SOLUTIONS
5. CHAPTER 2 TEST SOLUTIONS
3 LINEAR EQUATIONS
1. 3.1 Solving One-Step Equations
2. 3.2 Solving Two-Step Equations
3. 3.3 Translating Between Words and Symbols
4. 3.4 Solving Word Problems
5. 3.5 Equations with Like Terms
6. 3.6 Equations with Variables on Both Sides
7. 3.7 Equations with Parentheses
8. 3.8 Using Tables to Solve Word Problems Involving Values
9. 3.9 Transforming Formulas
10. CHAPTER 3 TEST
11. CHAPTER 3 SOLUTIONS
12. CHAPTER 3 TEST SOLUTIONS
4 RATIOS AND PROPORTIONS
1. 4.1 Expressing Ratios in Simplest Form
2. 4.2 Using Ratios in the Real World
3. 4.3 Proportions and Equations with Fractions
4. 4.4 Converting Units
5. 4.5 Percents
6. CHAPTER 4 TEST
7. CHAPTER 4 SOLUTIONS
8. CHAPTER 4 TEST SOLUTIONS
5 LINEAR INEQUALITIES
1. 5.1 Basic Principles of Solving Inequalities
2. 5.2 Representing Inequalities
3. 5.3 Solving Linear Inequalities
4. 5.4 Compound Inequalities
5. 5.5 Word Problems with Inequalities
6. CHAPTER 5 TEST
7. CHAPTER 5 SOLUTIONS
8. CHAPTER 5 TEST SOLUTIONS
6 FUNCTIONS AND GRAPHS WITH TWO VARIABLES
1. 6.1 Functions and Function Notation
2. 6.2 Introduction to Graphing
3. 6.3 Characteristics of Graphs
4. 6.4 Evaluating Functions from Equations and Graphs
5. CHAPTER 6 TEST
6. CHAPTER 6 SOLUTIONS
7. CHAPTER 6 TEST SOLUTIONS
7 LINEAR FUNCTIONS AND THEIR GRAPHS
1. 7.1 Introduction to Linear Functions
2. 7.2 Slope Formula
3. 7.3 Determining If an Ordered Pair Is a Solution to an Equation
4. 7.4 Writing the Equation of a Line
5. 7.5 Solving Systems of Linear Equations by Graphing
6. 7.6 Solving Systems of Linear Equations by Substitution
7. 7.7 Solving Systems of Linear Equations by Elimination
8. 7.8 Solving Systems of Linear Inequalities
9. 7.9 Word Problems with Systems of Linear Equations and Inequalities
10. CHAPTER 7 TEST
11. REVIEW TEST 1: CHAPTERS 1–7
12. CHAPTER 7 SOLUTIONS
13. CHAPTER 7 TEST SOLUTIONS
14. REVIEW TEST 1 SOLUTIONS
8 OPERATIONS WITH POLYNOMIALS
1. 8.1 Adding and Subtracting Polynomials
2. 8.2 Multiplying Monomials: Rules of Exponents
3. 8.3 Dividing Monomials: Rules of Exponents
4. 8.4 Multiplying Polynomials
5. 8.5 Dividing Polynomials
6. CHAPTER 8 TEST
7. CHAPTER 8 SOLUTIONS
8. CHAPTER 8 TEST SOLUTIONS
9 QUADRATIC FUNCTIONS
1. 9.1 Factoring a Monomial from a Polynomial
2. 9.2 Factoring by Grouping
3. 9.3 Factoring Trinomials
4. 9.4 Special Cases of Factoring
5. 9.5 Solving Quadratic Equations by Factoring
6. 9.6 Radical Expressions
7. 9.7 Solving Quadratic Equations by Completing the Square
8. 9.8 Solving Quadratic Equations Using the Quadratic Formula
9. 9.9 Graphing Quadratic Functions
10. 9.10 Solving Quadratic Equations by Graphing
11. 9.11 Solving Quadratic Equations by the Mean-Product Method
12. 9.12 Solving Quadratic-Linear Systems
13. 9.13 Using Quadratic Equations to Solve Word Problems
14. CHAPTER 9 TEST
15. CHAPTER 9 SOLUTIONS
16. CHAPTER 9 TEST SOLUTIONS
10 EXPONENTIAL FUNCTIONS
1. 10.1 Graphing Exponential Functions
2. 10.2 Using Exponential Functions to Solve Word Problems
3. CHAPTER 10 TEST
4. CHAPTER 10 SOLUTIONS
5. CHAPTER 10 TEST SOLUTIONS
11 SEQUENCES
1. 11.1 Writing Recursive Formulas for Sequences
2. 11.2 Writing Explicit Formulas for Arithmetic and Geometric Sequences
3. 11.3 Modeling with Sequences
4. CHAPTER 11 TEST
5. CHAPTER 11 SOLUTIONS
6. CHAPTER 11 TEST SOLUTIONS
12 SUMMARY OF FUNCTIONS
1. 12.1 Cubic, Square Root, and Cube Root Functions
2. 12.2 Piecewise Functions
3. 12.3 Transformations of Functions
4. 12.4 Average Rate of Change of Functions
5. 12.5 Comparing Functions
6. CHAPTER 12 TEST
7. CHAPTER 12 SOLUTIONS
8. CHAPTER 12 TEST SOLUTIONS
13 STATISTICS
1. 13.1 Two-Way Tables
2. 13.2 Dotplots and Histograms
3. 13.3 Shape, Center, and Spread
4. 13.4 Scatterplots and Regression
5. CHAPTER 13 TEST
6. REVIEW TEST 2: CHAPTERS 8–13
7. CHAPTER 13 SOLUTIONS
8. CHAPTER 13 TEST SOLUTIONS
9. REVIEW TEST 2 SOLUTIONS
FORMULAS
1. Linear Equations
2. Exponents
3. Radicals
4. Quadratic Equations
5. Exponential Equations
6. Sequences
7. Functions
8. Statistics
GLOSSARY OF MATHEMATICAL SYMBOLS
1. Basic Operations
2. Ratios and Proportions
3. Polynomial and Radical Expressions
4. Equations and Inequalities
5. Functions and Graphs
6. Statistics
GLOSSARY OF MATHEMATICAL TERMS
ABOUT THE AUTHORS
INDEX
END USER LICENSE AGREEMENT

List of Tables

Chapter 1
1. Table 1.1 Operations with signed numbers.
2. Table 1.2 Properties of Numbers
Chapter 2
1. Table 2.1 Operations with fractions.
Chapter 3
1. Table 3.1 Translating words into symbols.
2. Table 3.2 How to solve word problems.
3. Table 3.3 Examples and non-examples of like terms.
4. Table 3.4 Variables on 1 vs. 2 sides of an equation.
5. Table 3.5 Relating rate to distance and time.
Chapter 4
1. Table 4.1 Students in a 3:2 ratio.
2. Table 4.2 375 students in a 3:2 ratio.
3. Table 4.3 Problems with ratios or proportions.
4. Table 4.4 Place value.
5. Table 4.5 Problems with percentage.
Chapter 5
1. Table 5.1 Representing inequalities.
2. Table 5.2 Operations with inequalities.
3. Table 5.3 Inequalities in words.
Chapter 6
1. Table 6.1 Functional and non-functional vending machines.
2. Table 6.2 Characteristics of functions.
Chapter 7
1. Table 7.1 Types of slope.
2. Table 7.2 How to write the equation of a line.
3. Table 7.3 We can eliminate u or v.
4. Table 7.4 Comparing solution methods for systems of equations.
5. Table 7.5 Comparing systems of equations and inequalities.
Chapter 8
1. Table 8.1 Polynomials.
2. Table 8.2 Rules of exponents.
Chapter 9
1. Table 9.1 Vocabulary for multiplication, division, and factoring.
2. Table 9.2 Perfect squares.
3. Table 9.3 Examples and non-examples of radicals in simplest form.
4. Table 9.4 Operations with radicals.
5. Table 9.5 Examples of completing the square.
6. Table 9.6 Properties determined from equations of parabolas.
7. Table 9.7 Methods for Solving Quadratic Equations.
Chapter 10
1. Table 10.1 Number of grains of wheat per square.
Chapter 11
1. Table 11.1 Comparing subscript notation and function notation.
2. Table 11.2 Explicit formula for an arithmetic sequence.
3. Table 11.3 Explicit formula for a geometric sequence.
4. Table 11.4 Comparing recursive and explicit formulas.
Chapter 12
1. Table 12.1 Comparing quadratic and square root functions.
2. Table 12.2 Comparing cubic and cube root functions.
3. Table 12.3 Parent functions.
4. Table 12.4 Summary of transformation.
Chapter 13
1. Table 13.1 Frequency table.
2. Table 13.2 Measures of shape.
3. Table 13.3 Measures of center.
4. Table 13.4 Measures of spread.
5. Table 13.5 Hours studied vs. test grade.
6. Table 13.6 Form of a bivariate relationship.
7. Table 13.7 Direction of a bivariate relationship.
8. Table 13.8 Table with exponential growth.
9. Table 13.9 Scatterplots and residual plots.

List of Illustrations

Chapter 1
1. Figure 1.1 Number line
2. Figure 1.2 Terms associated with division.
3. Figure 1.3 Order of operations.
4. Figure 1.4 Sets of numbers.
Chapter 2
1. Figure 2.1 Dividing a rectangular pizza into eighths.
2. Figure 2.2 Numerator and denominator.
3. Figure 2.3 Adding fractions.
4. Figure 2.4 Multiplying fractions.
5. Figure 2.5 Adding unlike denominators.
Chapter 3
1. Figure 3.1 How many forks and how many plates?
2. Figure 3.2 Group similar items.
3. Figure 3.3 Vocabulary for like terms.
4. Figure 3.4 How many chocolate squares?
5. Figure 3.5 Distributive property.
6. Figure 3.6 Calculating coin value.
Chapter 4
1. Figure 4.1 Number line showing a 6:4 ratio.
2. Figure 4.2 5 students in a 3:2 ratio.
3. Figure 4.3 10 students in a 3:2 ratio.
4. Figure 4.4 15 students in a 3:2 ratio.
5. Figure 4.5 30 students in a 3:2 ratio.
6. Figure 4.6 Means and extremes in a proportion.
7. Figure 4.7 3%.
Chapter 5
1. Figure 5.1 5 is greater than 3.
2. Figure 5.2 x < 10.
3. Figure 5.3 x = 10.
4. Figure 5.4 x ≤ 10.
5. Figure 5.5 Interval notation for x ≤ 10.
6. Figure 5.6 Set-builder notation for x ≤ 10.
7. Figure 5.7 Graphing an intersection.
Chapter 6
1. Figure 6.1 Vocabulary for relations.
2. Figure 6.2 Function as a table, diagram, or set of ordered pairs.
3. Figure 6.3 Non-function as a table, diagram, or set of ordered pairs.
4. Figure 6.4 Vocabulary for the coordinate plane.
5. Figure 6.5 Graph of a table vs. graph of an equation.
Chapter 7
1. Figure 7.1 Describe the patterns in the tables.
2. Figure 7.2 Constant difference between consecutive y-values.
3. Figure 7.3 Three views of a function.
4. Figure 7.4 Slope of a line.
5. Figure 7.5 Point-slope form of the equation of a line.
6. Figure 7.6 System of linear equations.
7. Figure 7.7 Some systems aren't solved easily by graphing.
8. Figure 7.8 x is less than 10.
9. Figure 7.9 x is less than 10 on the coordinate plane.
Chapter 8
1. Figure 8.1 Multiplication as a rectangle.
2. Figure 8.2 Multiplication as addition of smaller rectangles.
3. Figure 8.3 Multiplication with the area model.
Chapter 9
1. Figure 9.1 Find the missing factor or product.
2. Figure 9.2 Find the missing factors.
3. Figure 9.3 Factoring binomials with a common binomial factor.
4. Figure 9.4 Factor by grouping.
5. Figure 9.5 Examples of multiplying binomials.
6. Figure 9.6 Factoring with the ac method.
7. Figure 9.7 Vocabulary for radicals.
8. Figure 9.8 Subsets of the real numbers.
9. Figure 9.9 Table and graph of a quadratic function.
10. Figure 9.10 Vocabulary for parabolas.
11. Figure 9.11 Mistakes to avoid when graphing parabolas.
12. Figure 9.12 Mean-product method.
Chapter 10
1. Figure 10.1 Equation of an exponential function.
2. Figure 10.2 Equation of an exponential function with a rate.
Chapter 11
1. Figure 11.1 Recursive formula for a sequence.
2. Figure 11.2 Comparing sequences and equations.
Chapter 12
1. Figure 12.1 Vocabulary for cubic functions.
2. Figure 12.2 Leading coefficients of cubic polynomials.
3. Figure 12.3 Piecewise function.
4. Figure 12.4 Vertical shift.
5. Figure 12.5 Horizontal shift.
6. Figure 12.6 Vertical stretch or compression.
7. Figure 12.7 Comparing the rate of change of linear and nonlinear functions....
8. Figure 12.8 Average rate of change.
Chapter 13
1. Figure 13.1 Bar graph.
2. Figure 13.2 Two-way table.
3. Figure 13.3 Joint frequency in a two-way table.
4. Figure 13.4 Bar graph vs. histogram.
5. Figure 13.5 The meaning of the mean.
6. Figure 13.6 Scatterplot of hours studied vs. test grade.
7. Figure 13.7 Strength of a bivariate relationship: correlation coefficient.
8. Figure 13.8 Residuals.

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Cover Image and Design: Paul McCarthy

INTRODUCTION

What is algebra? You may associate it with solving equations such as 2x + 7 = 19. However, both the history of algebra and the way that it's taught today show that algebra is much more. For thousands of years, people solved algebraic problems without symbols such as x and +. By the 9th century, people including the Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī had popularized the idea of using an algorithm (a set of well-defined instructions) to determine unknown quantities. In fact, the word algebra comes from the Arab word al-jabr, meaning “the reduction,” from the title of al-Khwārizmī's most famous mathematical text, Kitāb al-jabr wa al-muqābalah. Symbolic notation didn't become widespread until European mathematicians such as François Viète and René Descartes developed them in the 16th and 17th centuries. Nowadays, algebra courses include not just equations but also functions (the special rules that define mathematical relationships) and real-world modeling with statistics. In short, today's algebra students must know how to understand word problems, make and interpret graphs, create and solve equations, and draw appropriate conclusions from data.

Not surprisingly, algebra makes many people nervous. Maybe you recall endless drills and elaborate procedures from years ago. Perhaps you're a middle school or high school student who's intimidated by the high level of abstract reasoning that's required. If so, you're not alone. We understand how you feel! For many years, we've taught all levels of high school math, so we have a lot of experience working with diverse learners. This book contains concrete strategies that help our students succeed. We strongly believe that people can get better at math if they have access to the right tools.

We wrote this book as a general introduction to algebra. We assume that you're familiar with basic arithmetic (adding, subtracting, multiplying, and dividing numbers) and fractions. If you're not comfortable with these topics, don't worry—we briefly review them in Chapters 1 and 2. Even if you are comfortable with them, we suggest that you look through these chapters anyway. We explain why these ideas work and how they're related to the algebraic ideas we discuss later on.

Each chapter in this book is divided into sections, with model examples and tips. At the end of each section, you'll find several exercises to help you practice and apply your skills. These exercises include what we call Questions to Think About (open-ended questions designed to help you think about important concepts) as well as dozens of word problems. Each chapter has a test with multiple-choice and open-ended questions. The solutions to all exercises and chapter tests are located at the end of each chapter.

As you work through this book, you'll see some important ideas about algebra that we emphasize:

Algebra is a language. We believe that many people find algebra intimidating because the words and symbols we use, such as polynomial, a_n, and f(x), literally look like a different language. In addition, we don't just write math, we also read and speak it. In the Reading and Writing Tips, we discuss how to write and pronounce mathematical symbols as well as how to use them in context. We also include a glossary of mathematical terms and symbols in the back of the book.
Algebra should make sense. We believe that algebra should be taught in a way that makes sense. In our experience, part of the reason why so many people suffer from math anxiety is that they see it as a collection of disjointed and confusing tricks. Throughout this book, we use techniques (such as the area model for multiplication) that relate to other mathematical topics, such as geometry and statistics. By making these connections, you can extend what you learned in one situation to another context, which will strengthen your mathematical skills and boost your confidence!
Algebra requires pictures. As we taught during the pandemic, we had to adjust our instruction. We couldn't be with our students in person, so they often had to teach themselves more independently. Incorporating graphs, tables, diagrams, and other images into our teaching helped our students make sense of math. Since this book is a self-teaching guide, we've included many visual strategies throughout this book.
Algebra requires technology. Calculators, computers, and other technology aren't just shortcuts for menial computations. They are now required for today's complex modeling tasks. Using technology helps us to see patterns more efficiently. Since each of these tools has vastly different user instructions, we don't include specific instructions for each device. Instead, we include Technology Tips that apply no matter what device you're using.
Algebra is a human endeavor. We believe that algebra should not be perceived as a set of rigid rules developed by a select group of people. In fact, as we note throughout this book, many mathematical concepts were developed in different cultures around the world over thousands of years. (We mention some of the more interesting stories in the Did You Know? callouts.) In addition, we recognize that making mistakes is a natural part of doing math. In the Watch Out! callouts, we point out many of the common errors that we've seen students make over the years so that you can avoid them!

We hope that as you work through this book, you'll find that algebra can be less intimidating and more meaningful than you originally thought.

— Bobson Wong and Larisa Bukalov

Wiley Self-Teaching Guides teach practical skills from accounting to astronomy, management to mathematics. Look for them at your local bookstore.

Practical Algebra

A Self-Teaching Guide

ACKNOWLEDGMENTS

INTRODUCTION