Answer
$x\le-3 \text{ OR } x\ge13$
Work Step by Step
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the given inequality, $
|x-5|\ge8
,$ implies
\begin{array}{l}\require{cancel}
x-5\ge8
\\\\\text{OR}\\\\
x-5\le-8
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
x-5\ge8
\\
x-5+5\ge8+5
\\
x\ge13
\\\\\text{OR}\\\\
x-5\le-8
\\
x-5+5\le-8+5
\\
x\le-3
.\end{array}
Hence, the solution is $
x\le-3 \text{ OR } x\ge13
$.
The graph of the solution above is shown below.
