#### Answer

$y=\left\{ -\dfrac{106}{5}, -\dfrac{34}{5} \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
-5|y+14|=-36
,$ isolate first the absolute value expression. Then use the definition of absolute value equality. Do checking of the solution/s.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
-5|y+14|=-36
\\\\
\dfrac{-5|y+14|}{-5}=\dfrac{-36}{-5}
\\\\
|y+14|=\dfrac{36}{5}
.\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}
y+14=\dfrac{36}{5}
\\\\\text{OR}\\\\
y+14=-\dfrac{36}{5}
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
y+14=\dfrac{36}{5}
\\\\
y=\dfrac{36}{5}-14
\\\\
y=\dfrac{36}{5}-\dfrac{70}{5}
\\\\
y=-\dfrac{34}{5}
\\\\\text{OR}\\\\
y+14=-\dfrac{36}{5}
\\\\
y=-\dfrac{36}{5}-14
\\\\
y=-\dfrac{36}{5}-\dfrac{70}{5}
\\\\
y=-\dfrac{106}{5}
.\end{array}
If $y=-\dfrac{34}{5},$ then
\begin{array}{l}\require{cancel}
-5|y+14|=-36?
\\\\
-5\left| -\dfrac{34}{5}+14 \right|=-36?
\\\\
-5\left| -\dfrac{34}{5}+\dfrac{70}{5} \right|=-36?
\\\\
-5\left| \dfrac{36}{5} \right|=-36?
\\\\
-5\left( \dfrac{36}{5} \right)=-36?
\\\\
-36=-36
\text{ (TRUE)}
.\end{array}
If $y=-\dfrac{106}{5},$ then
\begin{array}{l}\require{cancel}
-5|y+14|=-36?
\\\\
-5\left| -\dfrac{106}{5}+14 \right|=-36?
\\\\
-5\left| -\dfrac{106}{5}+\dfrac{70}{5} \right|=-36?
\\\\
-5\left| -\dfrac{36}{5} \right|=-36?
\\\\
-5\left( \dfrac{36}{5} \right)=-36?
\\\\
-36=-36
\text{ (TRUE)}
.\end{array}
Hence, $
y=\left\{ -\dfrac{106}{5}, -\dfrac{34}{5} \right\}
.$