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 5, 5. Point B is at negative 4, negative 2. Point C is at 2, 1. Point D is at negative 2, 4. Point E is at 2, 4. Point F is at 3, negative 4.

Part A: Using the graph above, create a system of inequalities that only contain points D and E 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 E are solutions to the system of inequalities created in Part A. (3 points)

Part C: Timothy can only attend a school in his designated zone. Timothy's zone is defined by y < 3x − 3. Explain how you can identify the schools that Timothy is allowed to attend. (2 points)

3 answers

i really need help

Part A: To create a system of inequalities that only contain points D and E in the overlapping shaded regions, we need to determine the equations of the lines that form the boundary of the shaded regions.

1. We first need to find the equation of the line passing through points D and E. Using the slope-intercept form (y = mx + b), we can calculate the slope:

Slope = (y2 - y1) / (x2 - x1)
Slope = (4 - 2) / (-2 - 2)
Slope = 2 / -4
Slope = -0.5

We can choose either point D or E to substitute into the equation. Let's use point E(2, 4):
4 = -0.5 * 2 + b
4 = -1 + b
b = 5

So the equation of the line passing through points D and E is: y = -0.5x + 5.

2. Now we need to find the equation of the line that separates the regions containing points D and E. This line should pass through point C(2, 1) but should not include point C. So let's find the equation of the line using point-slope form:

y - y1 = m(x - x1)
y - 1 = -0.5(x - 2)
y - 1 = -0.5x + 1
y = -0.5x + 2

3. To ensure that only points D and E are in the overlapping shaded regions, we need to combine the inequalities for both lines. Let's call the line passing through points D and E as L1, and the line separating the regions as L2. The system of inequalities would be:

For L1: y ≤ -0.5x + 5
For L2: y > -0.5x + 2

To graph and shade the lines on the coordinate grid, plot the points and draw the lines corresponding to each inequality. For L1, draw a dashed line because it includes points D and E and shade the region below the line. For L2, draw a solid line since it excludes point C and shade the region above the line. The overlapping shaded region will indicate the area where only points D and E lie.

Part B: To verify that points D and E are solutions to the system of inequalities created in Part A, substitute the coordinates of each point into the corresponding inequalities and check if the inequalities hold true.

For Point D(-2, 4):
L1: 4 ≤ -0.5(-2) + 5
4 ≤ 1 + 5
4 ≤ 6 (True)

L2: 4 > -0.5(-2) + 2
4 > 1 + 2
4 > 3 (True)

For Point E(2, 4):
L1: 4 ≤ -0.5(2) + 5
4 ≤ -1 + 5
4 ≤ 4 (True)

L2: 4 > -0.5(2) + 2
4 > -1 + 2
4 > 1 (True)

Since both points satisfy the system of inequalities, they are verified as solutions.

Part C: To identify the schools that Timothy is allowed to attend, we need to substitute the coordinates of each school (points A to F) into the inequality y < 3x - 3. If the inequality holds true, the school is within Timothy's zone.

For Point A(-5, 5):
5 < 3(-5) - 3
5 < -15 - 3
5 < -18 (False)

For Point B(-4, -2):
-2 < 3(-4) - 3
-2 < -12 - 3
-2 < -15 (False)

For Point C(2, 1):
1 < 3(2) - 3
1 < 6 - 3
1 < 3 (True)

For Point D(-2, 4):
4 < 3(-2) - 3
4 < -6 - 3
4 < -9 (False)

For Point E(2, 4):
4 < 3(2) - 3
4 < 6 - 3
4 < 3 (False)

For Point F(3, -4):
-4 < 3(3) - 3
-4 < 9 - 3
-4 < 6 (True)

Based on the results, the schools that Timothy is allowed to attend are Point C and Point F.

@Damon

@Ms.sue