Answer
$\left( -\dfrac{9}{5},3 \right)$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|-5x+3| \lt 12
,$ use the definition of absolute value inequalities. Use the properties of inequalities to isolate the variable.
For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$
For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$
$\bf{\text{Solution Details:}}$
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{array}{l}\require{cancel}
-12 \lt -5x+3 \lt 12
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-12-3 \lt -5x+3-3 \lt 12-3
\\\\
-15 \lt -5x \lt 9
.\end{array}
Dividing by a negative number (and consequently reversing the sign), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-15}{-5} \gt \dfrac{-5x}{-5} \gt \dfrac{9}{-5}
\\\\
3 \gt x \gt -\dfrac{9}{5}
\\\\
-\dfrac{9}{5} \lt x \lt 3
.\end{array}
In interval notation, the solution set is $
\left( -\dfrac{9}{5},3 \right)
.$
The colored graph is the graph of the solution set.