Chapter 2 - Section 2.7 - Absolute Value Inequalities - Exercise Set - Page 106: 43

$\left( -\infty,-11 \right)$ OR $\left( 1,\infty \right) $

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.