Answer
$-\dfrac{7}{2}\lt a \lt 2$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|4a+3|\lt11
,$ use the definition of a less than (less than or equal to) absolute value inequality. Then use the properties of inequality to isolate the variable.
$\bf{\text{Solution Details:}}$
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}
-11\lt 4a+3 \lt11
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-11\lt 4a+3 \lt11
\\\\
-11-3\lt 4a+3-3 \lt11-3
\\\\
-14\lt 4a \lt 8
\\\\
-\dfrac{14}{4}\lt \dfrac{4a}{4} \lt \dfrac{8}{4}
\\\\
-\dfrac{7}{2}\lt a \lt 2
.\end{array}
Hence, the solution set $
-\dfrac{7}{2}\lt a \lt 2
.$
