#### Answer

$\left[ -5,9 \right]$

#### Work Step by Step

Using the properties of inequality, the given statement, $
|x-2|-3\le4
,$ is equivalent to
\begin{array}{l}\require{cancel}
|x-2|\le4+3
\\\\
|x-2|\le7
.\end{array}
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$, then the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-7\le x-2\le7
.\end{array}
Using properties of inequality, then
\begin{array}{l}\require{cancel}
-7\le x-2\le7
\\\\
-7+2\le x-2+2\le7+2
\\\\
-5\le x\le9
.\end{array}
Hence, the solution set is the interval $
\left[ -5,9 \right]
.$