To determine the intersection of the half-planes of the given inequalities, we can graph them on the same coordinate plane and check for overlapping regions. To do that, we have to identify their boundary lines, which we can do by isolating y in both inequalities.
x-y ≤ 4 & (I) x-y ≥ 4 & (II)
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(I), (II): Solve for y
(I), (II): LHS-x=RHS-x
- y ≤ - x+ 4 - y ≥ - x+ 4
(I), (II): Flip inequality and change signs
y ≥ x- 4 y ≤ x- 4
Looking at the inequalities, we can see that the only thing that separates them is the inequality symbol. This means they have the same boundary line!
y=x-4
This line has a slope of 1 and a y-intercept of -4. Let's graph this boundary line twice, a red line and a blue line, representing each of the inequalities.
To determine which region to shade, we will use (0,0) as a test point in each of the inequalities. If the inequality remains true after substituting the test point, we shade the side of the inequality that contains this test point. If not, we shade the opposite region.
y ≥ x- 4 & (I) y ≤ x- 4 & (II)
(I), (II): x= 0, y= 0
0 ? ≥ 0- 4 0 ? ≤ 0 - 4
(I), (II): Subtract term
0≥ - 4 ✓ 0 ≰ - 4 *
The results tells us that for (I) we shade the region above the boundary line, as it contains (0,0), and vice verse for (II).
From the graph above, we can see that the only region both solutions sets have in common is the boundary line. As such, the intersection is the boundary line.
y=x-4
