Answer
$\left( -5,11 \right)$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$, then the given inequality, $
|x-3|\lt8
,$ is equivalent to
\begin{array}{l}\require{cancel}
-8\lt x-3\lt8
\\\\
-8+3\lt x-3+3\lt8+3
\\\\
-5\lt x\lt11
.\end{array}
Hence, the solution set is $
\left( -5,11 \right)
.$
See graph below.