Elementary Algebra

Toby Wagner

Description

Elementary Algebra provides precollege algebra students with the essentials for understanding what algebra is, how it works, and why it so useful. It is written with plain language and includes annotated examples and practice exercises so that even students with an aversion to math will understand these ideas and learn how to apply them. This textbook expands on algebraic concepts that students need to progress with mathematics at the college level, including linear models and equations, polynomials, and quadratic equations. Written by faculty at Chemeketa Community College for the students in the classroom, Elementary Algebra is a classroom-tested textbook that sets students up for success.

Sujets

• Sciences et techniques
• Álgebra
• Algebra
• Mathematics
• Abrégé du calcul par la restauration et la comparaison
• Elementary

Informations

• Publié par: Chemeketa Press
• Date de parution: 01 mai 2021
• Nombre de lectures: 10
• EAN13: 9781943536894
• Langue: English
• Poids de l'ouvrage: 15 Mo

Extrait

Elementary Algebra
by Toby Wagner
Elementary Algebra
ISBN: 978-1-943536-29-0
Edition 1.3 Fall 2019
2019, Chemeketa Community College. All rights reserved.
Chemeketa Press
Chemeketa Press is a nonprofit publishing endeavor at Chemeketa Community College. Working together with faculty, staff, and students, we develop and publish affordable and effective alternatives to commercial textbooks. All proceeds from the sale of this book will be used to develop new textbooks. For more information, please visit chemeketapress.org .
Publisher: Tim Rogers
Managing Editor: Steve Richardson
Production Editor: Brian Mosher
Manuscript Editors: Steve Richardson, Matt Schmidgall
Design Editors: Ronald Cox IV, Kristen MacDonald
Cover Design: Ronald Cox IV, Faith Martinmaas, Shaun Jaquez, Kristi Etzel
Interior Design: Ronald Cox IV, Kristi Etzel
Layout: Noah Barrera, Matthew Sanchez, Faith Martinmaas, Emily Evans, Steve Richardson, Kristi Etzel, Cierra Maher, Candace Johnson
Additional contributions to the design and publication of this textbook come from the students and faculty in the Visual Communications program at Chemeketa.
Chemeketa Math Faculty
The development of this text and its accompanying MyOpenMath classroom has benefited from the contributions of many Chemeketa math faculty in addition to the author, including:
Ken Anderson, Lisa Healey, Kelsey Heater, Kyle Katsinis, Tim Merzenich, Nolan Mitchell, Chris Nord, Martin Prather, and Rick Rieman
Text Acknowledgment
This book was originally developed using materials from Elementary Algebra , by Wade Ellis and Dennis Burzynski, which has been made available under a Creative Commons Attribution 2.0 license and may be downloaded for free from legacy.cnx.org/content/col10614/1.3/ .
Printed in the United States of America.
Contents
Chapter 1: Solving Linear Equations and Inequalities
1.1 Solving Linear Equations
1.2 Solving Multi-Step Equations
1.3 Solving Literal Equations
1.4 Solving Linear Inequalities
Chapter 2: Linear Models
2.1 Introduction to Linear Models
2.2 Tables and Graphs
2.3 Slope
2.4 Slope-Intercept Form
2.5 Standard Form
2.6 Ordered Pairs and Equations
2.7 Line of Best Fit
2.8 Parallel and Perpendicular Lines
Chapter 3: Systems of Linear Equations
3.1 Using Graphing to Solve Systems
3.2 Using Substitution to Solve Systems
3.3 Using Elimination to Solve Systems
3.4 Using Systems to Solve Mixture Problems
Chapter 4: Polynomials
4.1 Addition and Subtraction of Polynomials
4.2 Multiplication of Polynomials
4.3 Rules of Exponents
4.4 Power Rules of Exponents
4.5 Negative Exponents
4.6 Applications of Exponent Rules
Chapter 5: Factoring
5.1 Factoring Out the Greatest Common Factor
5.2 Factoring Trinomials of the Form x 2 + bx + c
5.3 Special Cases for Factoring
5.4 Simplifying Rational Expressions
Chapter 6: Quadratic Equations
6.1 Solving Quadratic Equations by Factoring
6.2 Understanding and Simplifying Square Roots
6.3 Solving Quadratic Equations by Using Square Roots
6.4 Solving Quadratic Equations by Using the Quadratic Formula
6.5 Graphing Quadratic Equations
6.6 Solving Quadratic Equations by Graphing
Solutions to Odd-Numbered Exercises
Glossary/Index
CHAPTER 1
Solving Linear Equations and Inequalities
Solving equations is at the heart of algebra. Because of this, it seems fitting to begin our study of elementary algebra by learning how to solve equations algebraically. In Chapter 1 , we will focus on solving linear equations, which are typically less complicated than non-linear equations. We will also learn how to solve linear inequalities. As we progress through the chapter, we will be using our skills to help us solve application problems, which are commonly known as word problems or story problems.
In this chapter, you ll study the following topics:
1.1 Solving Linear Equations
1.2 Solving Multi-Step Equations
1.3 Solving Literal Equations
1.4 Solving Linear Inequalities

1.1 Solving Linear Equations
Overview
Here s a problem:

The area of a rectangle is the same as the value found by multiplying the length and the width. A rectangle that is 14 feet long has an area of 91 square feet. What is the width of the rectangle?
This problem states that two things are the same - the area of the rectangle and the value found by multiplying the length and width. Mathematicians write many sentences like this, though usually with mathematical notation instead of words. In math, equations are used to communicate sameness. Equations are the most common sentences in math.
When you are finished with this section, you will be able to:
Identify various types of equations
Understand the meaning of solutions and equivalent equations
Use addition and subtraction to solve 1-step equations
Use multiplication and division to solve 1-step equations
Solve application problems involving 1-step equations
At the end of this section, we will write and solve an equation to find the width of the rectangle in the problem above. In the meantime, let s learn more about equations.
A. Types of Equations
An equation is a mathematical sentence that asserts that two things are the same or equal . An equals sign ( ) means is the same as. It s important, though, to understand that an equation only asserts that two things are the same. As you ll soon see, this doesn t guarantee that the statement is actually true.
Some equations are always true. These equations are called identities . Identities are equations that are true for all acceptable values of the variable. Here are some examples of identities:
5 x 5 x
is true for all acceptable values of x because 5 times any number is always the same as 5 times that same number.
y 1 y 1
is true for all acceptable values of y because any number plus 1 is always the same as that same number plus 1.
2 5 7
is true, and no substitutions are necessary because this equation doesn t use any variables.
About Acceptable Values

For the equations in the first few chapters of this book, the variables can be replaced with any real number. All real numbers are therefore acceptable values of the variable. Later, you will see equations with more complicated expressions in which some numbers will not be used to replace the variable.
Some equations are never true. These equations are called contradictions . Contradictions are equations in which the expression on the left side and the expression on the right side are never equal, no matter what value is substituted for the variable.
x x 1
is never true for any acceptable value of x because no real number is equal to itself plus 1.
0 k 14
is never true for any acceptable value of k because the product of 0 and any real number is always 0. It can never be 14.
2 1
is never true because 2 can never equal 1.
The truth of some equations depends upon the number value chosen for the variable. These equations are called conditional equations . Conditional equations are true when using at least one replacement value for the variable and false when using at least one different replacement value for the variable.
x 6 11
is true only if x is replaced with 5 ( x 5). It is false if x is replaced with any other number.
y - 7 -1
is true only if y 6, and it s false if y is replaced with any other number.
A conditional equation with one variable is a linear equation if the highest power of the variable is 1. For example, t 5 12 is a linear equation because the only variable that appears is being raised to the power of 1 ( t t 1 ). However, the equation m 2 6 m 16 is not a linear equation because it contains a variable that is being raised to the power of 2. In Chapter 2 , we will see that if an equation is linear, then the graph of that equation is a straight line.
The following examples will show you different types of equations and explain why each is an identity, a contradiction, or conditional equation.

Example 1
Decide whether each equation is an identity, contradiction, or a conditional equation .
1. x - 4 6
2. n - 2 n - 2
3. a 5 a 1
Solutions
1. x 4 6 is a conditional equation because it s true only on the condition that x 10.
2. n 2 n 2 is an identity because it is true for all values of n . For example, if n 5, then 5 2 5 2 is true. And if n 7, then 7 2 7 2 is true. No matter what number we substitute for n , the equation will always be true.
3. a 5 a 1 is a contradiction because every value of a produces a false statement. For example, if a 8, then 8 5 8 1 is false. And if a 2, then 2 5 2 1 is false.
Practice A

Now it s your turn to classify a new set of equations. For each of the following equations, decide whether the equation is an identity, a contradiction, or a conditional equation . If you think that the equation is conditional, figure out the value of the variable that will make the equation true. When you are done, turn the page and check your solutions.
1. x 1 10
2. y 4 7
3. 5 a 25
4. = 9
5.
6. y 2 y 2
7. x 4 x 3
8. x x x 3 x
9. 8 x 0
10. m 7 5
B. Solutions
The equals sign of an equation indicates that the value represented by the expression to the left of the equals sign is the same as the value represented by the expression to the right of the equals sign.

Figure 1. Equivalent Equations
When we use variables like x or y , the values for those variables that will make an equation true are called solutions of the equation. An equation is solved when all of its solutions have been found.
Sometimes a set of equations will share the exact same solution. When that happens, the equations are called equivalent equations . For example, these equations are equivalent:
2 x 1 7 2 x 6 x 1 2
The only substituted value for x that makes each equation true is 3, so the solution for all three equations is the same, x 3.
Take a look at the table in Figure 1. You ll notice that 6 is the solution for all three equations. Tha