Answer
$(-\infty,-4]\cup(2,\infty)$.
The graph of the solution set is shown below.
Work Step by Step
The given compound inequality is
$x+3\leq-1$ or $-4x+3\lt-5$.
Solve each inequality separately.
First $=x+3\leq-1$.
Subtract $3$ from both sides.
$=x+3-3\leq-1-3$
Simplify.
$=x\leq-4$
Second $=-4x+3\lt-5$.
Subtract $3$ from both sides.
$=-4x+3-3\lt-5-3$
Simplify.
$=-4x\lt-8$
Divide both sides by $-4$ and change the sense of the inequality.
$=\frac{-4x}{-4}\gt\frac{-8}{-4}$
Simplify.
$=x\gt2$
First graph then take the union of the two inequality.
We can write the compound inequality.
$x\leq-4$ as $(-\infty,-4]$ or $x\gt2$ as $(2,\infty)$
The union is
$(-\infty,-4]\cup(2,\infty)$.
The combined graph is shown below.
