Answer
$-5\lt x \lt 9$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given inequality, $
|x-2|\lt7
,$ 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}
-7\lt x-2 \lt7
.\end{array}
Using the properties of inequality, the inequality above is equivalent to
\begin{array}{l}\require{cancel}
-7+2\lt x-2+2 \lt7+2
\\\\
-5\lt x \lt 9
.\end{array}
Hence, the solution set is $
-5\lt x \lt 9
.$
