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 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 A 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 A and E are solutions to the system of inequalities created in Part A. (3 points)

Part C: William can only attend a school in his designated zone. William's zone is defined by y < −x − 1. Explain how you can identify the schools that William is allowed to attend. (2 points)

Answer 1

Part A: You want two inequalities such that they overlap. So I'd draw two lines that slant downward at 45° below point C and another parallel line above point F. The shaded area should be between these to lines only.
First, we choose the lines...
y = -x + 2 has C and F above it and all other noted points below it. y = -x + 9 has all the points below it on the graph.
So if we include all points above the first line and all points below the second line, we obtain our intersected shaded area. To do this changes our linear equations to linear inequalities.
All points above y = -x + 2 would then be y > -x + 2
All points below y = -x + 9 would then be y < -x +9
Part B: Verify that these points satisfy the system of inequalities by substituting (x, y) for both points into both inequalities. [Pick any random point outside this shaded area to prove that it coordinates do not satisfy the system.]
Graph the line y = -2x + 2. That will be a line that has a y-intercept of 2 and a slope of -2. Then shade *below* that line (we shade below because it is y < ...). The schools that will be in the shaded region are B, A, and D.
Hope this helps-Maranda