a To graph the inequality, we have to draw the boundary line, then decide which region to shade.
B
b To graph the inequality, we have to draw the boundary line, then decide which region to shade.
C
c To graph the inequality, we have to draw the boundary line, then decide which region to shade.
D
d To graph the inequality, we have to draw the boundary line, then decide which region to shade.
A
a Graph C
B
b Graph A
C
c Graph D
D
d Graph B
Practice makes perfect
a To match the inequality with the correct graph, we should graph it. When graphing an inequality we first have to identify its boundary line. It can be found by replacing the inequality symbol with an equals sign.
Inequality: & 3x-2y ≤ 6
Boundary Line: & 3x-2y = 6
To draw this line, we will first rewrite the equation in slope-intercept form.
Now that the equation is in slope-intercept form, we can identify the slope m and y-intercept b.
y=3/2x-3 ⇔ y=3/2x+( - 3)
We will plot the y-intercept at (0, - 3), and then use the slope m=32 to plot another point on the line. Note that the boundary line must be solid because the inequality is not strict.
To decide which region of the coordinate plane to shade, we will substitute a test point that is not on the boundary line into the given inequality. If the substitution creates a true statement, we shade the region that includes the test point. Otherwise, we shade the opposite region. Let's use ( 0, 0) as our test point.
Since the substitution of the test point created a true statement, we will shade the region that contains the point.
The graph of the inequality matches with Graph C.
b Notice that we have almost the same inequality as in Part A. The only difference is that the inequality sign, this time, is strict. Therefore, the shaded region and the boundary line will be the same, but the boundary line will be dashed.
The graph of the inequality matches with Graph A.
c The boundary line is the same as in Parts A and B. However, since the inequality sign is reversed with respect to the previous parts, the shaded half-plane will be the region below the line. Since the inequality is strict, the line will be dashed.
The graph of the inequality matches with Graph D.
d Finally, this last inequality is almost the same as the one in Part C. The only difference is that this is a non-strict inequality. Therefore, the boundary line and the region will be the same as in Part C, but the line will be solid.
