#### Answer

$x=\{ -6,-1 \}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
|7+2x|=5
,$ use the definition of absolute value equality. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
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}
7+2x=5
\\\\\text{OR}\\\\
7+2x=-5
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
7+2x=5
\\\\
2x=5-7
\\\\
2x=-2
\\\\
x=-\dfrac{2}{2}
\\\\
x=-1
\\\\\text{OR}\\\\
7+2x=-5
\\\\
2x=-5-7
\\\\
2x=-12
\\\\
x=-\dfrac{12}{2}
\\\\
x=-6
.\end{array}
Hence, $
x=\{ -6,-1 \}
.$
The colored points is the graph of the solution set.