Answer
$m=\left\{ -2,12 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|5-m|+9=16
,$ isolate first the absolute value expression. Then use the definition of absolute value equality. Finally, use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the proeprties of equality to isolate the absolute value expression, the given is equivalent to
\begin{array}{l}\require{cancel}
|5-m|+9=16
\\\\
|5-m|=16-9
\\\\
|5-m|=7
.\end{array}
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}
5-m=7
\\\\\text{OR}\\\\
5-m=-7
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
5-m=7
\\\\
-m=7-5
\\\\
-m=2
\\\\
m=-2
\\\\\text{OR}\\\\
5-m=-7
\\\\
-m=-7-5
\\\\
-m=-12
\\\\
m=12
.\end{array}
Hence, $
m=\left\{ -2,12 \right\}
.$