Question

The coordinate plane below represents a city.

Points A through F are schools in the city. graph of coordinate plane. Point A is at negative 3, negative 4. Point B is at negative 4, 3. Point C is at 2, 2. Point D is at

The coordinate plane is below

Part A: Using the graph above, create a system of inequalities that only contain 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: Natalie can only attend a school in her designated zone. Natalie's zone is defined by y < −2x + 2. Explain how you can identify the schools that Natalie is allowed to attend.

Answer 1

Part A) The system of inequalities is

`x\geq2` and
`y\geq2`

Part B) In the procedure

Part C) The schools that Natalie is allowed to attend are A,B and D

Explanation:

Part A: Using the graph above, create a system of inequalities that only contain points C and F in the overlapping shaded regions

we have

Points C(2,2), F(3,4)

The system of inequalities could be

`x\geq2` -----> inequality A

The solution of the inequality A is the shaded area at the right of the solid line x=2

`y\geq2` -----> inequality B

The solution of the inequality B is the shaded area above of the solid line y=2

see the attached figure N 1

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

we know that

If a ordered pair is a solution of the system of inequalities, then the ordered pair must satisfy both inequalities

Verify point C

C(2,2)

Inequality A

`x\geq2` ----->
`2\geq2` ----> is true

Inequality B

`y\geq2` ------>
`2\geq2` ----> is true

therefore

Point C is a solution of the system of inequalities

Verify point D

F(3,4)

Inequality A

`x\geq2` ----->
`3\geq2` ----> is true

Inequality B

`y\geq2` ------>
`4\geq2` ----> is true

therefore

Point D is a solution of the system of inequalities

Part C: Natalie can only attend a school in her designated zone. Natalie's zone is defined by y < −2x + 2. Explain how you can identify the schools that Natalie is allowed to attend.

we have

`y < -2x+2`

The solution of the inequality is the shaded area below the dotted line
`y=-2x+2`

The y-intercept of the dotted line is the point (0,2)

The x-intercept of the dotted line is the point (1,0)

To graph the inequality, plot the intercepts and shade the area below the dotted line

see the attached figure N 2

therefore

The schools that Natalie is allowed to attend are A,B and D