Question

The coordinate plane below represents a town. Points A through F are farms in the town.

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 C and F in the overlapping shaded regions. Explain how the lines will be graphed and shaded on the coordinate grid above.

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

Part C: Chickens can only be raised in the area defined by y < 5x − 3. Explain how you can identify farms in which chickens can be raised.

Answer 1

we have
A( 2,-3) B(-3,-4)  C(-4,2)  D(2,4)  E(3,1)  F(-2,3)

using a graph tool
see the attached figure N 1

 
Part A;There are many systems of inequalities that can be created such that only contain points C and F in the overlapping shaded regions.
Any system of inequalities which is satisfied by (-4, 2) and (-2, 3) but is not satisfied by ( 2,-3) (-3,-4), (2, 4) and (3, 1) can serve.
An example of such system of equation is  
x < 0
y > 0
The system of the equation above represents all the points in the second quadrant of the coordinate system. The area above the x-axis and to the left of the y-axis is shaded.
see the attached figure N 2

Part B:
It can be verified that points C and F are solutions to the system of inequalities above by substituting the coordinates of points C and F into the system of equations and see whether they are true.
Substituting C(-4,2) into the system we have:
-4 < 0
2 > 0
as can be seen, the two inequalities above are true, hence point C is a solution to the set of inequalities.
Also, substituting F(-2,3) into the system we have
:-2 < 0
3 > 0
as can be seen, the two inequalities above are true, hence point F is a solution to the set of inequalities.

Part C:
Given that chicken can only be raised in the area defined by y < 5x - 3.
To identify the farms in which chicken can be raised, we substitute the coordinates of the points A to F into the inequality defining chicken's area.

For point A( 2,-3): -3 < 5(2) - 3 ⇒ -3 < 10 - 3 ⇒ -3 < 7 which is true
For point B(-3,-4): -4 < 5(-3) - 3 ⇒ -4 < -15 - 3 ⇒ -4 <-18 which is false
For point C(-4,2):  2 < 5(-4) - 3 ⇒ 2 < -20 - 3 ⇒ 2 <-23 which is false
For point D(2,4):  4 < 5(2) - 3 ⇒ 4 < 10 - 3 ⇒ 4 <7 which is true
For point E(3,1):  1 < 5(3) - 3 ⇒ 1 < 15 - 3 ⇒ 1 <12 which is true
For point F(-2,3):  3 < 5(-2) - 3 ⇒ 3 < -10 - 3 ⇒ 3 < -13 which is false

Therefore
 the farms in which chicken can be raised are the farms at point A, D,  and E

see the attached figure N 3.