Answer
$\left[ -2,8 \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|3-x| \le 5
,$ 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}
-5 \le 3-x \le 5
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-5-3 \le 3-x-3 \le 5-3
\\\\
-8 \le -x \le 2
.\end{array}
Dividing by a negative number (and consequently reversing the sign), the inequality above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{-8}{-1} \ge \dfrac{-x}{-1} \ge \dfrac{2}{-1}
\\\\
8 \ge x \ge -2
\\\\
-2 \le x \le 8
.\end{array}
In interval notation, the solution set is $
\left[ -2,8 \right]
.$
The colored graph is the graph of the solution set.