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.

Sketch of the region mod x < mod y