Answer
$-2\lt x \lt 6$
Refer to the graph below.
Work Step by Step
Using the properties of inequality, the given, $
|3x-6|+3\lt15
,$ is equivalent to
\begin{align*}\require{cancel}
|3x-6|+3-3&\lt15-3
\\
|3x-6|&\lt12
.\end{align*}
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 implies
\begin{align*}\require{cancel}
-12\lt 3x-6 &\lt12
.\end{align*}
Using the properties of inequality, the inequality above is equivalent to
\begin{align*}\require{cancel}
-12+6\lt 3x-6+6 &\lt12+6
\\
-6\lt 3x &\lt 18
\\\\
-\dfrac{6}{3}\lt \dfrac{3x}{3} &\lt \dfrac{18}{3}
\\\\
-2\lt x &\lt 6
.\end{align*}
Hence, the solution is $
-2\lt x \lt 6
.$
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 from $
-2
$ to $
6
$ with hollowed dots at $
-2
$ and $
6
$.
