Answer
$z\lt-4\text{ OR }z\gt4$
Refer to the graph below.
Work Step by Step
Using the properties of inequality, the given, $
|3z|-4\gt8
,$ is equivalent to
\begin{align*}\require{cancel}
|3z|-4&\gt8
\\
|3z|-4+4&\gt8+4
\\
|3z|&\gt12
.\end{align*}
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c \text{ or } x\lt-c$ (which is equivalent to $|x|\ge c$ implies $x\ge c \text{ or } x\le-c$), the inequality above implies
\begin{align*}\require{cancel} 3z&\gt12
\\\\&\text{OR}\\\\
3z&\lt-12
.\end{align*}
Using the properties of inequality, the inequality above is equivalent to
\begin{align*}\require{cancel}
3z&\gt12
\\
\dfrac{3z}{3}&\gt\dfrac{12}{3}
\\
z&\gt4
\\\\&\text{OR}\\\\
3z&\lt-12
\\
\dfrac{3z}{3}&\lt-\dfrac{12}{3}
\\
z&\lt-4
.\end{align*}
Hence, the solution is $
z\lt-4\text{ OR }z\gt4
.$
Since a hollowed dot is used for the symbols $\lt$ and $\gt,$ while a solid dot is used for the symbols $\le$ and $\ge,$ then the graph of the solution above is the set of numbers to the left of $
-4
$ and to the right of $
4
$ with hollowed dots at $
-4
$ and $
4
$.
