#### Answer

$-\dfrac{13}{2} \lt a \lt \dfrac{5}{2}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
30-4|a+2|\gt12
,$ isolate first the absolute value expression. Then use the definition of a less than (less than or equal to) absolute value inequality. Use the properties of inequality to isolate the variable. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality to isolate the absolute value expression results to
\begin{array}{l}\require{cancel}
30-4|a+2|\gt12
\\\\
-4|a+2|\gt12-30
\\\\
-4|a+2|\gt-18
.\end{array}
Dividing both sides by a negative number (and consequently reversing the inequality symbol), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-4|a+2|\gt-18
\\\\
|a+2|\lt\dfrac{-18}{-4}
\\\\
|a+2|\lt\dfrac{9}{2}
.\end{array}
Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-\dfrac{9}{2}\lt a+2 \lt\dfrac{9}{2}
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-\dfrac{9}{2}\lt a+2 \lt\dfrac{9}{2}
\\\\
-\dfrac{9}{2}-2\lt a+2-2 \lt\dfrac{9}{2}-2
\\\\
-\dfrac{9}{2}-\dfrac{4}{2}\lt a \lt\dfrac{9}{2}-\dfrac{4}{2}
\\\\
-\dfrac{13}{2} \lt a \lt \dfrac{5}{2}
.\end{array}
Hence, the solution set $
-\dfrac{13}{2} \lt a \lt \dfrac{5}{2}
.$