Review question

# Where is $|x|$ less than $|y|$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6392

## Solution

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.