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)

I don't see any graph ...

Part A: To create a system of inequalities that contain only points D and E in the overlapping shaded regions, we need to examine the graph above.

First, let's identify the lines that form the boundaries of the shaded regions. There are two lines: the blue line passing through points (0, 3) and (3, 0), and the red line passing through points (0, -2) and (2, 0).

The blue line can be written as an equation using the slope-intercept form: y = -x + 3.
To ensure that only points D and E are included in the shaded region below this line, we should use the inequality y < -x + 3.

The red line can also be written as an equation using the slope-intercept form: y = -x - 2.
To ensure that only points D and E are included in the shaded region below this line, we should use the inequality y < -x - 2.

So, the system of inequalities that contain only points D and E in the overlapping shaded regions is:
y < -x + 3
y < -x - 2

To graph and shade these lines on the coordinate grid, we would draw both lines y = -x + 3 and y = -x - 2 using their respective slopes. The lines would be dotted since the inequalities are strict inequalities (y <). Then, we would shade the area below both lines, including only points D and E.

Part B: To verify if points D and E are solutions to the system of inequalities, we substitute their coordinates into the inequalities and check if the inequalities hold true.

Point D has coordinates (2, 1).
Substituting the values into the inequalities, we have:
1 < -2 + 3 (for y < -x + 3)
1 < -2 - 2 (for y < -x - 2)

Both of these inequalities are true, so point D is a solution to the system.

Point E has coordinates (1, 2).
Substituting the values into the inequalities, we have:
2 < -1 + 3 (for y < -x + 3)
2 < -1 - 2 (for y < -x - 2)

Both of these inequalities are also true, so point E is a solution to the system.

Therefore, both points D and E are solutions to the system of inequalities.

Part C: To identify the schools that Timothy can attend, we need to examine the inequality y < 3x - 3 that defines his designated zone.

The inequality y < 3x - 3 represents the equation of a line in slope-intercept form, where the slope is 3 and the y-intercept is -3. This line has a positive slope, indicating that it moves upward from left to right on the coordinate grid.

To identify the schools Timothy is allowed to attend, we can observe the region below the line. All the schools that lie below the line and within Timothy's designated zone of y < 3x - 3 are the ones he can attend.

In summary, to determine the schools that Timothy is allowed to attend, plot the line y = 3x - 3 on the coordinate grid and shade the region below it. The schools falling within this shaded region are the ones Timothy can attend.