To understand this problem, let's first define a half-plane. A half-plane is a part of the plane that lies on one side of a straight line and extends infinitely far in that direction. In this problem, we have an inequality that defines a half-plane.
The inequality is: 4x + 3y ≥ 6
To graph this inequality, we can first plot the line 4x + 3y = 6. To plot this line, we can find two points on the line by setting x = 0 and y = 0 and solving for the other variable.
When x = 0, we get: 3y = 6, y = 2
When y = 0, we get: 4x = 6, x = 3/2
So the two points on the line are (0, 2) and (3/2, 0). We can plot these points and draw a straight line passing through them.
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Now we need to determine which side of the line represents the half-plane defined by the inequality 4x + 3y ≥ 6.
To do this, we can choose a test point not on the line, such as the origin (0,0), and substitute its coordinates into the inequality:
4(0) + 3(0) ≥ 6
0 ≥ 6
Since this is false, the point (0,0) is not in the half-plane defined by the inequality. Therefore, we shade the half-plane that does not include the origin:
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The shaded portion shows the outer boundary of the half-plane defined by the inequality 4x + 3y ≥ 6.
Answer: 4x + 3y ≥ 6.
