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