Answer
$0\lt y\lt18$
Refer to the graph below.
Work Step by Step
Using the properties of inequality, the given, $
3|y-9|\lt27
,$ is equivalent to
\begin{align*}\require{cancel}
\dfrac{3|y-9|}{3}&\lt\dfrac{27}{3}
\\\\
|y-9|&\lt9
.\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 is equivalent to
\begin{align*}
-9\lt y-9&\lt9
.\end{align*}
Using the properties of inequality, the inequality above is equivalent to
\begin{align*}
-9+9\lt y-9+9&\lt9+9
\\
0\lt y&\lt18
.\end{align*}
Hence, the solution is $
0\lt y\lt18
.$
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 $0$ to $18$ with hollowed dots at $0$ and $18$.
