Answer
$4\le x \le10$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|x-7|\le3
,$ use the definition of absolute value inequality. Then use the properties of inequality to isolate the variable. Finally, graph the solution set.
In the graph, a hollowed dot is used for $\lt$ or $\gt.$ A solid dot is used for $\le$ 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 x-7 \le3
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-3\le x-7 \le3
\\\\
-3+7\le x-7+7 \le3+7
\\\\
4\le x \le10
.\end{array}
Hence, the solution set is $
4\le x \le10
.$
