Answer
$-4 \lt t \le -\dfrac{10}{3}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
4 \gt g(t) \ge 2
,$ replace the function with the given, $
g(t)=-3t-8
.$ Then use the properties of inequality to isolate the variable. Finally, graph the solution set.
$\bf{\text{Solution Details:}}$
Replacing the inequality with the given function, then
\begin{array}{l}\require{cancel}
4 \gt -3t-8 \ge 2
.\end{array}
Using the properties of inequality, the given is equivalent to
\begin{array}{l}\require{cancel}
4 \gt -3t-8 \ge 2
\\\\
4+8 \gt -3t-8+8 \ge 2+8
\\\\
12 \gt -3t \ge 10
.\end{array}
Multiplying both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
12 \gt -3t \ge 10
\\\\
\dfrac{12}{-3} \gt \dfrac{-3t}{-3} \ge \dfrac{10}{-3}
\\\\
-4 \lt t \le -\dfrac{10}{3}
.\end{array}
The graph includes the points from $-4$ (exclusive) to $-\dfrac{10}{3}$ (inclusive).
