Graph each inequality separately. The overlapping region will be the solution of the system.
Practice makes perfect
To solve the given system by graphing, we should first draw each inequality separately. Then we will combine the graphs. The overlapping region will be the solution set. Let's start!
Inequality I
To graph |y|≥ x, we will have to create a compound inequality first. Since |y| is greater than or equal to x, this will be an or compound inequality.
|y|≥ x ⇔ y≥ x or y≤- x
Note that in this case the solution will be a union of the solution sets of each inequality. Let's graph the cases one at a time.
Case 1
First, we will graph the inequality y≥ x. By exchanging the inequality symbol to an equals sign, we can find the boundary line y=x. We can tell that it passes through the origin and has a slope of 1. Also, the inequality is not strict, so the boundary line will be solid.
To decide which region we should shade, we can use a test point that does not lie on the boundary line. If it satisfies the inequality, it lies in the solution set. If not, we will shade the other region. Let's use ( 2, 1).
Because (2,1) did not create a true statement, we will shade the region opposite this point.
Case 2
For the second case we will follow the same process. The boundary line of this inequality is y=- x, so we will graph a solid line passing through the origin with the slope of -1.
Once more we shade the region opposite the test point.
Combining the Cases
Finally, we can combine the graphs of both cases. Remember that this time the solution set is a union. Thus, we will combine the shaded regions on one coordinate plane to form the graph of |y|≥ x.
Inequality II
Let's follow the same process for the second inequality, y<2x. The boundary line is y=2x, so we will plot a line passing through the origin with the slope of 2. Note that the inequality is strict, so the line will be dashed.
Let's use the point ( 2, 1) to decide which region to shade.
