Answer
$a=\left\{ -8,24 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
3|a-8|-4=44
,$ isolate first the absolute value expression. Then use the definition of absolute value equality. Do checking of the solution/s.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
3|a-8|=44+4
\\\\
3|a-8|=48
\\\\
|a-8|=\dfrac{48}{3}
\\\\
|a-8|=16
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
a-8=16
\\\\\text{OR}\\\\
a-8=-16
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
a-8=16
\\\\
a=16+8
\\\\
a=24
\\\\\text{OR}\\\\
a-8=-16
\\\\
a=-16+8
\\\\
a=-8
.\end{array}
If $a=24,$ then
\begin{array}{l}\require{cancel}
3|a-8|-4=44?
\\\\
3|24-8|-4=44?
\\\\
3|16|-4=44?
\\\\
3(16)-4=44?
\\\\
48-4=44
\\\\
44=44
\text{ (TRUE)}
.\end{array}
If $a=-8,$ then
\begin{array}{l}\require{cancel}
3|a-8|-4=44?
\\\\
3|-8-8|-4=44?
\\\\
3|-16|-4=44?
\\\\
3(16)-4=44?
\\\\
48-4=44?
\\\\
44=44
\text{ (TRUE)}
.\end{array}
Hence, $
a=\left\{ -8,24 \right\}
.$
