Illustrate by means of a sketch the subset of the \(x\text{-}y\) plane given by
\[\begin{equation*}
\{(x,y) : |x| < |y|\}.
\end{equation*}\]

There are four quadrants to consider.

- In the first quadrant, both \(x\) and \(y\) are positive so the inequality becomes \(x < y\).
- In the second, \(x\) is negative so \(|x|=-x\) and we have \(-x < y\).
- In the third, it becomes \(-x < -y\), i.e. \(x > y\).
- In the fourth, \(x < -y\), i.e. \(y < -x\).

We thus have the following sketch. The shaded regions satisfy the inequality. The dotted lines indicate that these are not included within the defined region.