Answer
$\left(
-\infty,-11
\right)$ OR $\left(
1,\infty
\right)
$
Work Step by Step
Using the definition of absolute value inequality, then
\begin{align*}
|x+5|+2&\ge8\\
|x+5|\ge6\\
x+5&\ge6\\
x&\ge1
\\\\\text{OR}\\\\
x+5&\le-6\\
x&\le-11
.\end{align*}
Hence, the interval that satisfies the given inequality is $\left(
-\infty,-11
\right)$ OR $\left(
1,\infty
\right)
$.
See attached file for the graph.