Answer
$1 \le x \le 4$
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).
