Answer
Interval: $(- \infty,\ -\frac{1}{2})\cup\ (\frac{1}{3},\ \infty)$
Work Step by Step
$\frac{1}{2}|4x+\frac{1}{3}|\gt\frac{5}{6}$
$|4x+\frac{1}{3}|\gt\frac{5*2}{6}$
$|4x+\frac{1}{3}|\gt\frac{5}{3}$
We split the absolute value into the positive and negative:
$4x+ \frac{1}{3}\gt\frac{5}{3}$
$ 4x\gt\frac{4}{3}$
$ x\gt\frac{1}{3}$
$4x+ \frac{1}{3}\lt-\frac{5}{3}$
$4x\lt-\frac{5}{3}-\frac{1}{3}$
$4x\lt-2$
$x\lt-\frac{1}{2}$
Interval: $(- \infty,\ -\frac{1}{2})\cup\ (\frac{1}{3},\ \infty)$