Answer
$(-\infty,-2)$.
The graph of the solution set is shown below.
Work Step by Step
The given compound inequality is
$\Rightarrow 5x-2\leq-22$ or $-3x-2\gt4$.
First
$\Rightarrow 5x-2\leq-22$
Add $2$ to both sides.
$\Rightarrow 5x-2+2\leq-22+2$
Simplify.
$\Rightarrow 5x\leq-20$
Divide both sides by $5$.
$\Rightarrow \frac{5x}{5}\leq\frac{-20}{5}$
Simplify.
$\Rightarrow x\leq-4$
Second
$\Rightarrow -3x-2\gt4$
Add $2$ to both sides.
$\Rightarrow -3x-2+2\gt4+2$
Simplify.
$\Rightarrow -3x\gt6$
Divide both sides by $-3$ and change the sense of the inequality..
$\Rightarrow \frac{-3x}{-3}\lt\frac{6}{-3}$
Simplify.
$\Rightarrow x\lt-2$
First graph then take the union of the solution sets of the two inequalities.
We can write the compound inequality.
$x\leq-4$ as $(-\infty,-4]$ or $x\lt -2$ as $(-\infty,-2)$
The union is
$(-\infty,-4]\cup(-\infty,-2)=(-\infty,-2)$.
The graph is shown below.
