A60

Appendix A

Review of Fundamental Concepts of Algebra

A.6

Linear Inequalities in One Variable

Introduction

Simple inequalities were discussed in Appendix A.1. There, you used the inequality symbols , and ≥ to compare two numbers and to denote subsets of real numbers. For instance, the simple inequality x ≥ 3 denotes all real numbers x that are greater than or equal to 3. Now, you will expand your work with inequalities to include more involved statements such as 5x and 3 ≤ 6x 1 < 3. 7 < 3x 9

What you should learn

• Represent solutions of linear inequalities in one variable. • Solve linear inequalities in one variable. • Solve inequalities involving absolute values. • Use inequalities to model and solve real-life problems.

Why you should learn it

Inequalities can be used to model and solve real-life problems. For instance, in Exercise 101 on page A68, you will use a linear inequality to analyze the average salary for elementary school teachers.

As with an equation, you solve an inequality in the variable x by finding all values of x for which the inequality is true. Such values are solutions and are said to satisfy the inequality. The set of all real numbers that are solutions of an inequality is the solution set of the inequality. For instance, the solution set of x 1 < 4

is all real numbers that are less than 3. The set of all points on the real number line that represent the solution set is the graph of the inequality. Graphs of many types of inequalities consist of intervals on the real number line. See Appendix A.1 to review the nine basic types of intervals on the real number line. Note that each type of interval can be classified as bounded or unbounded.

Example 1

Intervals and Inequalities

Write an inequality to represent each interval, and state whether the interval is bounded or unbounded. a. b. d. a. b. d. 3, 5 3, , 3, 5 corresponds to 3, , corresponds to corresponds to 3 < x ≤ 5. 3 < x.

< x <

c. 0, 2...