Question

JIMMMMMM

Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:

y ≥ −3x + 3
y is less than 3 over 2 times x minus 6

Part A: Describe the graph of the system, including shading and the types of lines graphed. Provide a description of the solution area. (6 points)

Part B: Is the point (−6, 3) included in the solution area for the system? Justify your answer mathematically. (4 points)

Answer 1

>= means greater than or equal to
<= means less than or equal to

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Part A

The graph of y >= -3x+3 will have a solid boundary line and the shading will be above the boundary line.

The boundary line y = -3x+3 has a negative slope so it moves down as you read it from left to right. It goes through the points (0,3) and (1,0)

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The graph of y < (3/2)x - 6 will have a dashed or dotted boundary line. The shading is below the boundary.

The graph y = (3/2)x-6 goes through the two points (0,-6) and (2,-3)

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If you graph both y >= -3x+3 and y < (3/2)x - 6 together, you get what you see in the attached image. This solution shaded region is the result of the overlapping prior shaded regions.

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Part B

Plug (x,y) = (-6,3) into each inequality to see if we get a true inequality or not

For the first inequality we have
y >= -3x+3
3 >= -3*(-6)+3
3 >= 18+3
3 >= 21
which is false. The value 3 is not larger or equal to 21. So right off the bat we know that (-6,3) is NOT a solution. It is NOT in the solution region.

Let's check the other inequality just for the sake of completeness
y < (3/2)x - 6
3 < (3/2)*(-6) - 6
3 < -9 - 6
3 < -15
this is also false. The value -15 is smaller than 3, since it is to the left of 3

We're given more evidence that (-6,3) is NOT in the solution area. It is outside of both shaded areas.

Answer 2

The point (-6,3) is not included in the solution area for the system.

Explanation:

The given inequities are

`y\geq -3x+3` .... (1)

`y<(3)/(2)x-3` .... (2)

Part A: The related equations are

`y=-3x+3`

`y=(3)/(2)x-3`

Put x=0 in the above equation to find the y-intercept and put y=0, to find the x-intercept.

`y=-3(0)+3=3`

The y-intercept of first equation is 3.

`0=-3x+3\Rightarrow x=1`

The y-intercept of first equation is 1.

`y=(3)/(2)(0)-3=-3`

The y-intercept of second equation is -3.

`0=(3)/(2)x-3\Rightarrow 2`

The y-intercept of first equation is 2.

The related line of inequality (1) is a solid line because the sign of inequality is ≥ and the related line of inequality (2) is a dotted line because the sign of inequality is <.

Shaded portion of first inequality is above the related line and the shaded portion of second inequality is below the related line.

The solution area is shown in the below graph.

Part B: The point (−6, 3) is not included in the solution area for the system.

Check the given inequalities by the point (-6,3).

`3\geq -3(6)+3`

`3\geq -18+3`

`3\geq -15`

This statement is false.

Therefore the point (-6,3) is not included in the solution area for the system.