Graph the system of inequalities presented here on your own paper, then use your graph to answer the following questions:y > 2x + 3y is less than negative 3 over 2 times x minus 4Part 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 (â’4, 6) included in the solution area for the system? Justify your answer mathematically. (4 points)

The x and y-intercepts of first line are (0,3) and (-1.5,0) respectively. The x and y-intercepts of second line are (0,-4) and (-2.667,0) respectively.

Check the given inequality by (0,0).

[tex]0>2(0)+3[/tex]

[tex]0>3[/tex]

This statement is false. It means (0,0) is not the solution of this inequality and the shaded region is opposite side of the origin.

[tex]0<-\frac{3}{2}(0)-4[/tex]

[tex]0<-4[/tex]

This statement is false. It means (0,0) is not the solution of this inequality and the shaded region is opposite side of the origin.

The graph of given inequalities is shown below.

Part B:

We have to check whether (4,6) is a solution of the system of inequality or not.

Check the given inequality for the point (4,6).

[tex](6)>2(4)+3[/tex]

[tex]6>11[/tex]

This statement is false. It means (4,6) is not the solution of given system of inequalities.

wagonbelleville wagonbelleville · Answer 2 · 2019-02-11T12:41:26+01:00

Answer:

We are given the system of inequalities,

[tex]y>2x+3[/tex] and [tex]y<-\frac{3}{2}x-4[/tex]

Part A: We have that the equations given are straight line which have the intersection point (-2,-1).

According to 'Zero Test' which states that 'After substituting (0,0) in the inequalities. If the result is true, the solution region is towards the origin. If the result is false, the solution region is away from the origin'.

So, from the graph we see that the solution region is away from the origin, which means both the given inequalities fails the 'Zero Test'.

i.e. [tex]y>2x+3[/tex] ⇒ 0 > 3, which is false and [tex]y<-\frac{3}{2}x-4[/tex] ⇒ 0 < -4, which is also false.

Part B: We see that from the graph, that the point (4,6) lies outside the solution region.

Mathematically, we will substitute the value of x= 4 in the equality to check whether the value of y= 6.

So, [tex]y=2x+3[/tex] ⇒ [tex]y=2\times 4+3[/tex] ⇒ [tex]y=8+3[/tex] ⇒ y= 11

And, [tex]y=-\frac{3}{2}x-4[/tex] ⇒ [tex]y<-\frac{3}{2}\times 4-4[/tex] ⇒ [tex]y=-3\times 2-4[/tex] ⇒ [tex]y=-6-4[/tex] ⇒ y= -10.

Hence, we see that mathematically the point (4,6) does not lie in the solution region.