Answer
$\left( -2,\dfrac{1}{2} \right)$
Work Step by Step
Using the properties of equality, the given expression, $
-9+|3+4x|\lt-4
,$ is equivalent to
\begin{array}{l}\require{cancel}
|3+4x|\lt-4+9
\\\\
|3+4x|\lt5
.\end{array}
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a$, then the expression, $
|3+4x|\lt5
,$ is equivalent to
\begin{array}{l}\require{cancel}
-5\lt 3+4x\lt5
\\\\
-5-3\lt 3+4x-3\lt5-3
\\\\
-8\lt 4x\lt2
\\\\
-2\lt x\lt\dfrac{1}{2}
.\end{array}
Hence, the solution set is $
\left( -2,\dfrac{1}{2} \right)
.$
