Answer
$h=\left\{ \dfrac{5}{2},\dfrac{7}{2} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
8|h-3|=4
,$ 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}
8|h-3|=4
\\\\
|h-3|=\dfrac{4}{8}
\\\\
|h-3|=\dfrac{1}{2}
.\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}
h-3=\dfrac{1}{2}
\\\\\text{OR}\\\\
h-3=-\dfrac{1}{2}
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
h-3=\dfrac{1}{2}
\\\\
h=\dfrac{1}{2}+3
\\\\
h=\dfrac{1}{2}+\dfrac{6}{2}
\\\\
h=\dfrac{7}{2}
\\\\\text{OR}\\\\
h-3=-\dfrac{1}{2}
\\\\
h=-\dfrac{1}{2}+3
\\\\
h=-\dfrac{1}{2}+\dfrac{6}{2}
\\\\
h=\dfrac{5}{2}
.\end{array}
If $h=\dfrac{7}{2},$ then
\begin{array}{l}\require{cancel}
8|h-3|=4?
\\\\
8\left| \dfrac{7}{2}-3 \right|=4?
\\\\
8\left| \dfrac{7}{2}-\dfrac{6}{2} \right|=4?
\\\\
8\left| \dfrac{1}{2} \right|=4?
\\\\
8\left( \dfrac{1}{2} \right)=4?
\\\\
4=4
\text{ (TRUE)}
.\end{array}
If $h=\dfrac{5}{2},$ then
\begin{array}{l}\require{cancel}
8|h-3|=4?
\\\\
8\left| \dfrac{5}{2}-3 \right|=4?
\\\\
8\left| \dfrac{5}{2}-\dfrac{6}{2} \right|=4?
\\\\
8\left| -\dfrac{1}{2} \right|=4?
\\\\
8\left( \dfrac{1}{2} \right)=4?
\\\\
4=4
\text{ (TRUE)}
.\end{array}
Hence, $
h=\left\{ \dfrac{5}{2},\dfrac{7}{2} \right\}
.$
