Answer
$x=\left\{ -4,1 \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|3x+2|=|x-6|
,$ use the definition of an absolute equality. Then use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Since $|x|=|y|$ implies $x=y \text{ or } x=-y,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x+2=x-6
\\\\\text{OR}\\\\
3x+2=-(x-6)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x+2=x-6
\\\\
3x-x=-6-2
\\\\
2x=-8
\\\\
x=-\dfrac{8}{2}
\\\\
x=-4
\\\\\text{OR}\\\\
3x+2=-(x-6)
\\\\
3x+2=-x+6
\\\\
3x+x=6-2
\\\\
4x=4
\\\\
x=\dfrac{4}{4}
\\\\
x=1
.\end{array}
Hence, $
x=\left\{ -4,1 \right\}
.$