Answer
$-\dfrac{7}{2}\le w\le\dfrac{1}{2}$
Work Step by Step
Using the properties of inequality, the given, $
4|2w+3|-7\le9
,$ is equivalent to
\begin{array}{l}\require{cancel}
4|2w+3|-7+7\le9+7
\\
4|2w+3|\le16
\\\\
\dfrac{4|2w+3|}{4}\le\dfrac{16}{4}
\\\\
|2w+3|\le4
.\end{array}
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 implies
\begin{array}{l}\require{cancel}
-4\le2w+3\le4
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-4-3\le2w+3-3\le4-3
\\
-7\le2w\le1
\\\\
-\dfrac{7}{2}\le\dfrac{2w}{2}\le\dfrac{1}{2}
\\\\
-\dfrac{7}{2}\le w\le\dfrac{1}{2}
.\end{array}
Hence, the solution is $
-\dfrac{7}{2}\le w\le\dfrac{1}{2}
.$
The graph of the solution above is shown below.
