Answer:
Explanation:
The inequality 5x+12y ≤ 30 represents a shaded region in the xy-plane that includes all the points that satisfy the inequality. To graph this inequality, we can first graph the equation 5x+12y = 30, which is the boundary line of the shaded region.
To graph the boundary line, we can find its x- and y-intercepts:
x-intercept: Set y = 0 and solve for x: 5x + 12(0) = 30, which gives x = 6.
y-intercept: Set x = 0 and solve for y: 5(0) + 12y = 30, which gives y = 2.5.
So the boundary line passes through the points (6, 0) and (0, 2.5).
Next, we can determine which side of the boundary line to shade. One way to do this is to pick a test point that is not on the line and plug it into the inequality. For example, the point (0,0) is a convenient test point. Plugging in x=0 and y=0 into the inequality, we get:
5(0) + 12(0) ≤ 30
which simplifies to 0 ≤ 30. Since this is true, we know that the region containing the point (0,0) is part of the shaded region, and therefore the shaded region is the one that contains the origin.
Putting it all together, we can graph the inequality by drawing the line 5x+12y = 30 and shading the region below the line, as shown in the figure below:
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3 | x
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1 | /
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0 | /
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0 6
The shaded region includes all the points below the line 5x+12y=30, including the points on the line.