Answer
$\left[ -\dfrac{13}{3},\dfrac{11}{3} \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
-3 \le \dfrac{3x+1}{4} \le 3
,$ use the properties of inequality.
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 $\ge.$
$\bf{\text{Solution Details:}}$
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
4(-3) \le 4\left(\dfrac{3x+1}{4}\right) \le 4(3)
\\\\
-12 \le 3x+1 \le 12
\\\\
-12-1 \le 3x+1-1 \le 12-1
\\\\
-13 \le 3x \le 11
\\\\
-\dfrac{13}{3} \le \dfrac{3x}{3} \le \dfrac{11}{3}
\\\\
-\dfrac{13}{3} \le x \le \dfrac{11}{3}
.\end{array}
In interval notation, the solution set is $
\left[ -\dfrac{13}{3},\dfrac{11}{3} \right]
.$
The red graph is the graph of the solution set.