Answer
$x=\left\{ -1,9 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|x-4|=5
,$ use the definition of an absolute value equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to
\begin{array}{l}\require{cancel}
x-4=5
\\\\\text{OR}\\\\
x-4=-5
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-4=5
\\\\
x=5+4
\\\\
x=9
\\\\\text{OR}\\\\
x-4=-5
\\\\
x=-5+4
\\\\
x=-1
.\end{array}
Hence, $
x=\left\{ -1,9 \right\}
.$