#### Answer

$\color{blue}{\left\{-11, 3\right\}}$

#### Work Step by Step

RECALL:
$|x|=a \longrightarrow x=a \text{ or } x = -a$
Use the rule above to have:
$|x+4| = 7\longrightarrow x+4=7 \text{ or } x+4 = -7$
Solve each equation to obtain:
$x+4=7
\\x=7-4
\\x=3$
or
$x+4=-7
\\x=-7-4
\\x=-11$
Thus, the solution set is $\color{blue}{\left\{-11, 3\right\}}$.