Answer
$(-\infty,-5]\cup[1,\infty)$.
The graph is shown below.
Work Step by Step
The given expression is
$\Rightarrow -4|x+2|+5\leq-7$
Subtract $5$ from both sides.
$\Rightarrow -4|x+2|+5-5\leq-7-5$
Simplify.
$\Rightarrow -4|x+2|\leq-12$
Divide both sides by $-4$ and change the sense of the inequality.
$\Rightarrow \frac{-4|x+2|}{-4}\geq\frac{-12}{-4}$
Simplify.
$\Rightarrow |x+2|\geq3$
Rewrite the inequality without absolute value bars.
$\Rightarrow x+2\leq-3$ or $x+2\geq3$
Subtract $2$ from all three parts.
$\Rightarrow x+2-2\leq-3-2$ or $x+2-2\geq3-2$
Simplify.
$\Rightarrow x\leq-5$ or $x\geq1$
The solution set is less than or equal to $-5$ or greater than or equal to $1$.
The interval notation is
$(-\infty,-5]\cup[1,\infty)$.
