Problem: Draw the region of points {\displaystyle{(x,y)}} in the plane, which satisfy {\displaystyle{|y| {\le} |x| {\le} 1}}.

Solution: {\displaystyle{|y| {\le} |x| {\le} 1}} region will bounded by lines {\displaystyle{x = y}}, {\displaystyle{x = -y}}, {\displaystyle{x = -1}} & {\displaystyle{x = 1}}. Why is that?

First note that \( |x| \le 1 \) implies:

Inequality region

Similarly, if we demand \( |y| \le 1 \) (the double shaded zone).

Now if we want \( |y| \le |x| \) . This can be achieved by

  • \( y \le x \) when x and y are both positive (in the first quadrant); that is the region below the line x = y
  • \( y \le -x\) when x is negative and y positive (in the second quadrant); hence the region below the line y = -x
  • \(-y \le -x \) when (x, y) is in the third quadrant.
  • \( -y \le x \) when (x, y) is in fourth quadrant.

Therefore the final region is the following shaded region:

Inequality region