Question

The coordinate plane below represents a town. Points A through F are farms in the town. graph of coordinate plane. Point A is at 2, negative 3. Point B is at negative 3, negative 4. Point C is at negative 4, 2. Point D is at 2, 4. Point E is at 3, 1. Point F is at negative 2, 3. Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above. (5 points) Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A. (3 points) Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised. (2 points)

Answer 1

I made my own graph using the given points. Pls. see attachment.

Part A: Using the graph above, create a system of inequalities that only contains points D and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.

I used paint program to graph the points based on the given data.
I used a ruler as guide as I draw my line containing both points D and F and extended the line outward.
The yellow lines above represents the shaded portion of the graph.

y = mx + b

b is the y-intercept. It is the value of y when x is 0. In this case, b = 3.5
m is the slope. It is the change in y divided by the change in x.
point D (2,4) ; point F (-2,3)
change in y: 4 - 3 = 1
change in x: 2 - (-2) = 4
m = 1/4

y = 1/4 x + 3.5

Since we are looking for a system of inequality and the portion shaded is in the upper part of the line, the value of y will be greater than or equal to.

y ≥ 1/4x + 3.5

Part B: Explain how to verify that the points D and F are solutions to the system of inequalities created in Part A.

To check the solution given above. We will use the x values of points D and F and check whether its y values are equal to or greater than the solution.

y ≥ 1/4x + 3.5

Point D (2,4) : x = 2 : y ≥ 1/4 * 2 + 3.5 → 0.5 + 3.5 → 4
Point F (-2.3): x = -2 ; y ≥ 1/4 * -2 + 3.5 → -0.5 + 3.5 → 3

Part C: Chickens can only be raised in the area defined by y > 2x − 2. Explain how you can identify farms in which chickens can be raised.

y > 2x - 2

We will get the coordinates of each point and see if its x value will generate the desired value of y.

A(2,-3) ; x = 2 : y > 2(2) - 2 → 4 - 2 → 2 . DID NOT PASS. -3 is lesser than 2.
B(-3,-4) ; x = -3 : y > 2(-3) - 2 → -6 - 2 → -8. PASSED. -4 is greater than -8.
C(-4,2) ; x = -4 : y > 2(-4) - 2 → -8 - 2 → -10. PASSED. 2 is greater than -10.
D(2,4) ; x = 2 : y > 2(2) - 2 → 4 - 2 → 2. PASSED. 4 is greater than 2.
E(3,1) ; x = 3 : y > 2(3) - 2 → 6 - 2 → 4. DID NOT PASS. 1 is lesser than 4.
F(-2,3) ; x = -2 ; y > 2(-2) - 2 → -4 - 2 → -6. PASSED. 3 is greater than -6.

Among the farms, points B,C, D, and F can raise chickens based on the given system of linear inequality.