he 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)
Part A:
To create a system of inequalities that only contains points D and E in the overlapping shaded regions, we need to find two inequalities that restrict the region to these points.
Looking at the graph, we can see that both points D and E lie above the x-axis. So, the first inequality can be x ≥ 0. This ensures that both points D and E are on or to the right of the y-axis.
Now, let's focus on the y-values of points D and E. Point D has a y-value of 4, while point E has a y-value of 4 as well. Since these two points have the same y-value, we can set a second inequality that restricts the y-values to be no greater than 4. This inequality can be y ≤ 4.
Together, the system of inequalities that only contain points D and E in the overlapping shaded regions is:
x ≥ 0
y ≤ 4
To graph these inequalities, draw a solid vertical line at x = 0 to represent x ≥ 0. Then, draw a solid horizontal line at y = 4 to represent y ≤ 4. The shaded region where these two lines intersect will contain points D and E.
Part B:
To verify that points D and E are solutions to the system of inequalities created in Part A, we need to substitute the coordinate values of D and E into the inequalities and check if they satisfy the inequalities.
For point D (-2, 4):
Substituting x = -2 and y = 4 into the inequalities:
-2 ≥ 0 is false (not greater than or equal to 0)
4 ≤ 4 is true (equal to 4)
Point D satisfies only one of the inequalities, so it is not a solution to the system.
For point E (2, 4):
Substituting x = 2 and y = 4 into the inequalities:
2 ≥ 0 is true (greater than or equal to 0)
4 ≤ 4 is true (equal to 4)
Point E satisfies both inequalities and is a solution to the system.
Therefore, only point E is a solution to the system of inequalities in Part A.
Part C:
To identify the schools that Timothy is allowed to attend based on the inequality y < 3x - 3, we need to find the points that satisfy this inequality.
Let's consider the inequality y < 3x - 3. This inequality represents a boundary line on the coordinate plane. To graph this line, first, we draw a dashed line with a slope of 3 and a y-intercept of -3.
After graphing the line, we need to determine which side of the line satisfies the inequality y < 3x - 3. Since the inequality is strict (less than), we shade the region below the line (excluding the line itself). This shaded region represents the points that satisfy the inequality.
Any school in the shaded region below the line y < 3x - 3 is within Timothy's designated zone. Timothy is allowed to attend any school within this shaded region.