Answer
$\left( 1,9 \right)$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a,$ then the given inequality, $
|5-x|\lt4
,$ is equivalent to
\begin{array}{l}\require{cancel}
-4\lt 5-x\lt4
.\end{array}
Using the properties of inequality, then
\begin{array}{l}\require{cancel}
-4-5\lt 5-5-x\lt4-5
\\\\
-9\lt -x\lt-1
\\\\
9\gt x\gt1
\\\\
1\lt x\lt9
.\end{array}
Hence, the solution is the interval $
\left( 1,9 \right)
.$
