Toby Wagner - 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.

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 , 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, youll study the following topics:

Heres 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, lets learn more about 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.

Its important, though, to understand that an equation only asserts that two things are the same. As youll soon see, this doesnt 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.

About Acceptable Values For the equations in the first few chapters of this book, the variables can be replaced with any real number. 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.

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.

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 doesnt use any variables.
0 k = 14	is never true for any acceptable value of k because the product of 0 and any real number is always 0.
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.

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 its 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 = t1). However, the equation m2 + 6m = 16 is not a linear equation because it contains a variable that is being raised to the power of 2. In , 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. x 4 = 6 n 2 = n 2 a + 5 = a + 1 Solutions x 4 = 6 is a conditional equation because its true only on the condition that x = 10. 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. 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 its your turn to classify a new set of equations. For each of the following equations, decide whether the equation is an