## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

$-\dfrac{7}{2} \lt x \le 7$
$\bf{\text{Solution Outline:}}$ 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).