Answer
$-\dfrac{7}{2} \lt x \le 7 $
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
-21 \le f(x) \lt 0
,$ replace the function with the given, $
f(x)=-2x-7
.$ 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}
-21 \le -2x-7 \lt 0
.\end{array}
Using the properties of inequality, the given is equivalent to
\begin{array}{l}\require{cancel}
-21 \le -2x-7 \lt 0
\\\\
-21+7 \le -2x-7+7 \lt 0+7
\\\\
-14 \le -2x \lt 7
.\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}
-14 \le -2x \lt 7
\\\\
\dfrac{-14}{-2} \le \dfrac{-2x}{-2} \lt \dfrac{7}{{-2}}
\\\\
7 \ge x \gt -\dfrac{7}{2}
\\\\
-\dfrac{7}{2} \lt x \le 7
.\end{array}
The graph includes the points from $-\dfrac{7}{2}$ (exclusive) to $7$ (inclusive).
