Answer
$\left[ -\dfrac{10}{3},4 \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|3x-1| \le 11
,$ 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}
-11 \le 3x-1 \le 11
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-11+1 \le 3x-1+1 \le 11+1
\\\\
-10 \le 3x \le 12
\\\\
-\dfrac{10}{3} \le \dfrac{3x}{3} \le \dfrac{12}{3}
\\\\
-\dfrac{10}{3} \le x \le 4
.\end{array}
In interval notation, the solution set is $
\left[ -\dfrac{10}{3},4 \right]
.$
The colored graph is the graph of the solution set.