Answer
$r=\left\{ -3,-\dfrac{1}{3} \right\}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
-2|3r+5|+15=7
,$ isolate first the absolute value expression. Then use the definition of absolute value to analyze the solution.
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the given is equivalent to
\begin{array}{l}\require{cancel}
-2|3r+5|+15=7
\\\\
-2|3r+5|=7-15
\\\\
-2|3r+5|=-8
\\\\
|3r+5|=\dfrac{-8}{-2}
\\\\
|3r+5|=4
.\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}
3r+5=4
\\\\\text{OR}\\\\
3r+5=-4
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3r+5=4
\\\\
3r=4-5
\\\\
3r=-1
\\\\
r=-\dfrac{1}{3}
\\\\\text{OR}\\\\
3r+5=-4
\\\\
3r=-4-5
\\\\
3r=-9
\\\\
r=-\dfrac{9}{3}
\\\\
r=-3
.\end{array}
If $r=-\dfrac{1}{3},$ then
\begin{array}{l}\require{cancel}
-2|3r+5|+15=7?
\\\\
-2\left| 3\left( -\dfrac{1}{3} \right)+5 \right|+15=7?
\\\\
-2\left| -1+5 \right|+15=7?
\\\\
-2\left| 4 \right|+15=7?
\\\\
-2(4)+15=7?
\\\\
-8+15=7?
\\\\
7=7
\text{ (TRUE)}
.\end{array}
If $r=-3,$ then
\begin{array}{l}\require{cancel}
-2|3r+5|+15=7?
\\\\
-2\left| 3\left( -3 \right)+5 \right|+15=7?
\\\\
-2\left| -9+5 \right|+15=7?
\\\\
-2\left| -4 \right|+15=7?
\\\\
-2(4)+15=7?
\\\\
-8+15=7?
\\\\
7=7
\text{ (TRUE)}
.\end{array}
Hence, $
r=\left\{ -3,-\dfrac{1}{3} \right\}
.$