Answer
$x=\left\{ -7,1 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Given that $
f(x)=|2x+6|
,$ to find $x$ for which $
f(x)=8
,$ use substitution. Then use the definition of an absolute value equality. Finally, use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Replacing $f(x)$ with $
8
,$ then
\begin{array}{l}\require{cancel}
f(x)=|2x+6|
\\\\
8=|2x+6|
\\\\
|2x+6|=8
.\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}
2x+6=8
\\\\\text{OR}\\\\
2x+6=-8
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
2x+6=8
\\\\
2x=8-6
\\\\
2x=2
\\\\
x=\dfrac{2}{2}
\\\\
x=1
\\\\\text{OR}\\\\
2x+6=-8
\\\\
2x=-8-6
\\\\
2x=-14
\\\\
x=-\dfrac{14}{2}
\\\\
x=-7
.\end{array}
Hence, $
x=\left\{ -7,1 \right\}
.$
