The given function |y|>|x| is a "bowtie" inequality. For the "bowtie" inequality, there are four cases depending on its domain and range. Let's examine each case! In some cases, we might need to isolate y by multiplying both sides by -1. For those cases, remember that we should flip the inequality signs.
Case
|y| > |x|
Isolating y
I.
x≥ 0, y≥ 0
y> x
y > x
II.
x≤ 0, y≥ 0
y > - x
y > - x
III.
x≤ 0, y≤ 0
- y > - x
y < x
IV.
x≥ 0, y≤ 0
- y > x
y < - x
To graph the cases, we will determine their boundary lines by replacing the inequality signs with the equals sign.
Case
|y| > |x|
Isolating y
Boundary Line
I.
x≥ 0, y≥ 0
y > x
y > x
y = x
II.
x≤ 0, y≥ 0
y > - x
y > - x
y = - x
III.
x≤ 0, y≤ 0
- y > - x
y < x
y = x
IV.
x≥ 0, y≤ 0
- y > x
y < - x
y = - x
As we can see, there are two different boundary lines and they are both in slope-intercept form. Therefore, we can graph them by plotting their y-intercepts. Since the inequalities are strict, the boundary lines will be dashed.
Notice that Cases I, II, III, and IV represent Quadrants I, II, III, and IV, respectively. For the first case, the inequality states the points that have y-coordinates greater than x-coordinates. Thus, we will shade above the line.
Continuing in the same way, let's graph each case and complete the graph of the "bowtie."
