Shading half of the plane to show the solution set
Boundary Line
To graph the inequality, we have to draw the boundary line. The equation of a boundary line is found by replacing the inequality symbol from the inequality with an equals sign.
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Inequality &
Boundary Line [0.5em]
y ≤ 2x-3 &
y = 2x-3
Fortunately, this equation is already in slope-intercept form, so we can identify the slope m and y-intercept (0, b).
y = 2x-3 ⇔ y= 2x+( -3)
We will plot the y-intercept (0, -3), then use the slope m= 2 to plot another point on the line. Connecting these points with a solid line will give us the boundary line of our inequality. Notice that the boundary line is solid, not dashed, because the inequality is notstrict.
Shading the Plane
To decide which side of the boundary line 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 did not create a true statement, we will shade the region that does not contain the point.
Comparing The Graphs
Now that we have the graph of our inequality, let's compare it with the given one.
As we can see, it seems that the shaded region by our inequality is exactly the complement of the region shaded in the given diagram. It is correct as any point (x, y) either satisfies y > 2x - 3, either y ≤ 2x-3 — there is no other possibility. Notice that the boundary line also changed from dashed to solid.
