Answer
$-9\le x \le6$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|2x+3|\le15
,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-15\le 2x+3 \le15
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-15\le 2x+3 \le15
\\\\
-15-3\le 2x+3-3 \le15-3
\\\\
-18\le 2x \le12
\\\\
-\dfrac{18}{2}\le \dfrac{2x}{2} \le\dfrac{12}{2}
\\\\
-9\le x \le6
.\end{array}
Hence, the solution set is $
-9\le x \le6
.$
