Answer
$\left[ 2,8 \right]$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|5-x| \le 3
,$ 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}
-3 \le 5-x \le 3
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-3-5 \le 5-x-5 \le 3-5
\\\\
-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.