Answer
$\text{the interval }
\left( 3,7 \right)
$
Work Step by Step
Since for any nonnegative number $c$, $|x|\lt c$ implies $-c \lt x \lt c$, then the solution of the given inequality, $
|x-5|-4\lt-2
,$ is
\begin{array}{l}\require{cancel}
|x-5|\lt-2+4
\\\\
|x-5|\lt2
\\\\
-2\lt x-5\lt2
\\\\
-2+5\lt x-5+5\lt2+5
\\\\
3\lt x\lt7
.\end{array}
In interval notation, the solution is $
\text{the interval }
\left( 3,7 \right)
.$
