Answer
$\left[ -1,8 \right]$
Work Step by Step
Using the properties of equality, the given expression, $
-15+|2x-7|\le-6
,$ is equivalent to
\begin{array}{l}\require{cancel}
|2x-7|\le-6+15
\\\\
|2x-7|\le9
.\end{array}
Since for any $a\gt0$, $|x|\le a$ implies $-a\le x\le a$, then the expression, $
|2x-7|\le9
,$ is equivalent to
\begin{array}{l}\require{cancel}
-9\le 2x-7\le9
\\\\
-9+7\le 2x-7+7\le9+7
\\\\
-2\le 2x\le16
\\\\
-\dfrac{2}{2}\le \dfrac{2}{2}x\le\dfrac{16}{2}
\\\\
-1\le x\le8
.\end{array}
Hence, the solution set is $
\left[ -1,8 \right]
.$
See the graph below.