How do you graph the absolute value of x is less than or equal to the absolute value of y?

Answer 1

Break it up into the four quadrants.
Quadrant 1 (x ≥ 0, y ≥ 0): |x| = x, |y| = y, so we have x ≤ y <=> y ≥ x.
Quadrant 2 (x ≤ 0, y ≥ 0): |x| = -x, |y| = y, so we have -x ≤ y <=> y ≥ -x.
Quadrant 3 (x ≤ 0, y ≤ 0): |x| = -x, |y| = -y, so we have -x ≤ -y <=> y ≤ x.
Quadrant 4 (x ≥ 0, y ≤ 0): |x| = x, |y| = -y, so we have x ≤ -y <=> y ≤ -x.

So draw the lines y=x and y=-x. Shade above y=x in the first quadrant and below it in the third. Shade above y=-x in the second quadrant and below it in the fourth.

Another way to think of this is to say that |y| > k (for k positive) => y > k or y < -k, so |y| ≥ |x| <=> y ≥ |x| or y < -|x|. So draw y = |x| and y = -|x|, and shade above y = |x| and below y = -|x|. This is equivalent to the above method.

Answer 2

Draw a solid line at the slope and shade the numbers below it