#### Answer

Interval: $(-\infty,\ -9]\ \cup\ [7,\ \infty)$

#### Work Step by Step

$|\displaystyle \frac{x+1}{2}|\geq 4$
$\frac{1}{2}|x+1|\geq 4$
$|x+1|\geq 2*4$
$|x+1|\geq 8$
We split the absolute value into the positive and negative:
$x+1\geq 8$
$x\geq 7$
$x+1\leq-8$
$x\leq-9$
Interval: $(-\infty,\ -9]\ \cup\ [7,\ \infty)$