332 CHAPTER 6 Algebra, Graphs, and Functions We can determine the solution to an inequality by adding, subtracting, multiplying, or dividing both sides of the inequality by the same number or expression. We use the procedure discussed in Section 6.1 to isolate the variable, with one important exception: When both sides of an inequality are multiplied or divided by a negative number, the direction of the inequality symbol is reversed. Example 3 Multiplying by a Negative Number Solve the inequality x 3 − > and graph the solution set on the number line. Solution To solve this inequality, we must eliminate the negative sign in front of the x. To do so, we multiply both sides of the inequality by 1− and change the direction of the inequality symbol. x x x 3 1( ) 1(3) 3 − > − − < − < − Multiply both sides of the inequality by −1 and change the direction of the inequality symbol. The solution set is graphed on the number line as follows. 24 23 22 21 0 28 27 26 25 1 2 x < 23 7 Now try Exercise 19 Timely Tip Remember to change the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number. Example 4 Dividing by a Negative Number Solve the inequality x6 18 − < and graph the solution set on the number line. Solution To solve the inequality, we need to divide both sides of the inequality by the coefficient of the x-term, 6. − When dividing both sides of the inequality by 6, − change the direction of the inequality symbol because we are dividing by a negative number. x x x 6 18 6 6 18 6 3 − < − − > − > − Divide both sides of the inequality by 6− and change the direction of the inequality symbol. The solution set is graphed on the number line as follows. 23 22 21 0 1 27 26 25 24 2 3 4 x . 23 7 Now try Exercise 17 Example 5 Solving an Inequality Solve the inequality x3 5 7 − > and graph the solution set on the number line. Solution To determine the solution set, isolate x on one side of the inequality. x x x x x 3 5 7 3 5 5 7 5 3 12 3 3 12 3 4 − > − + > + > > > Add 5 to both sides of the inequality. Divide both sides of the inequality by 3. Learning Catalytics Keyword: Angel-SOM-6.5 (See Preface for additional details.)
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