#### Answer

$x=\left\{ -8,2 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|x-7|=|2x+1|
,$ 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}
x-7=2x+1
\\\\\text{OR}\\\\
x-7=-(2x+1)
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-7=2x+1
\\\\
x-2x=1+7
\\\\
-x=8
\\\\
x=-8
\\\\\text{OR}\\\\x-7=-(2x+1)
\\\\
x-7=-2x-1
\\\\
x+2x=-1+7
\\\\
3x=6
\\\\
x=\dfrac{6}{3}
\\\\
x=2
.\end{array}
Hence, $
x=\left\{ -8,2 \right\}
.$