# Chapter 9 - Inequalities and Problem Solving - 9.2 Intersections, Unions, and Compound Inequalities - 9.2 Exercise Set - Page 590: 66

$1 \le x \le 4$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $7 \ge g(x) \ge -2 ,$ replace $g(x)$ with the given function, $g(x)=3x-5 .$ Then use the properties of inequality to isolate the variable. Finally, graph the solution set. $\bf{\text{Solution Details:}}$ Replacing $g(x)$ with the given function, then \begin{array}{l}\require{cancel} 7 \ge 3x-5 \ge -2 .\end{array} Using the properties of inequality, the given is equivalent to \begin{array}{l}\require{cancel} 7+5 \ge 3x-5+5 \ge -2+5 \\\\ 12 \ge 3x \ge 3 \\\\ \dfrac{12}{3} \ge \dfrac{3x}{3} \ge \dfrac{3}{3} \\\\ 4 \ge x \ge 1 \\\\ 1 \le x \le 4 .\end{array} The graph includes the points from $1$ (inclusive) to $4$ (inclusive).

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