Answer
$\left[ -1,\dfrac{9}{5} \right]$
Work Step by Step
Using the properties of inequality, the given expression, $
-3+|5x-2|\le4
,$ is equivalent to
\begin{array}{l}\require{cancel}
|5x-2|\le4+3
\\\\
|5x-2|\le7
.\end{array}
Since for any $a\gt0$, $|x|\le a$ implies $-a\le x\le a$, then the given inequality, $
|5x-2|\le7
,$ is equivalent to
\begin{array}{l}\require{cancel}
-7\le 5x-2\le7
\\\\
-7+2\le 5x-2+2\le7+2
\\\\
-5\le 5x\le9
\\\\
-\dfrac{5}{5}\le \dfrac{5}{5}x\le\dfrac{9}{5}
\\\\
-1\le x\le\dfrac{9}{5}
.\end{array}
Hence, the solution set is $
\left[ -1,\dfrac{9}{5} \right]
.$
See graph below.