Answer
$-2\lt x \lt4$
Work Step by Step
$\bf{\text{Solution Outline:}}$
Solve the given inequality, $
|x-1|\lt3
,$ using the definition of a less than 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 given inequality is equivalent to
\begin{array}{l}\require{cancel}
-3\lt x-1 \lt3
.\end{array}
Using the properties of inequality to isolate the variable results to
\begin{array}{l}\require{cancel}
-3+1\lt x-1+1 \lt3+1
\\\\
-2\lt x \lt4
.\end{array}
Hence, the solution set is $
-2\lt x \lt4
.$
