Example 2. Find the range of $x$, if $|x-1|>2$.

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Solution. By the definition of absolute value

\[
\begin{gathered}
x-1>2 \text { when }(x-1)>0, \text { i.e., } x>1 \\
-(x-1)>2 \text { when }(x-1)<0, \text { i.e., } x<1
\end{gathered}
\]

From $x-1>2$ and $-x+1>2$ we get $x>3, x<-1$.

Therefore, the domain of the variable $x$ is the union of two infinite intervals: $(-\infty,-1),(3,+\infty)$.