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 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)

Answer 1

For part a,
y < x
y > -2x
Point B: (3,1) plugin x=3, y=1 y < x 1 < 3
which is true, so point B satisfies the first inequalityYou can graph the inequality y > 2x+5 and see which schools are in the shaded region

Answer 2

One possible answer is:

y ≤ 4 and
y > 1/2x.

Explanation:

The y-coordinate of both points D and E is 4.  This means if we specify that y≤4, anything above these points (point A) will not be included in the shaded area.  To graph this inequality, we draw a solid line at y=4 and shade the area below it.

When we graph the ordered pairs, we can see that F would be below the line that goes from B to C.  To find the equation of this line, we first find the slope using the formula 
$m=\frac{y_2-y_1}{x_2-x_1}$.

Using the coordinates of B and C, we have
$m=\frac{-2-1}{-4-2}=\frac{-3}{-6}=\frac{1}{2}$

Now we plug this into point-slope form,
y - y₁ = m(x - x₁)
y - y₁ = 1/2(x - x₁)

Using B as (x₁, y₁) we have
y - -2 = 1/2(x - -4)
y + 2 = 1/2(x + 4)

Using the distributive property,
y + 2 = 1/2*x + 1/2(4)
y + 2 = 1/2x + 2

Subtracting 2 from each side,
y + 2 - 2 = 1/2x + 2 - 2
y = 1/2x

This line goes through B and C.  We want the area above this, but we do not want B and C included; this means we specify that y is greater than this line:
 y > 1/2x

This would be graphed as a dotted line with the area above it shaded, up to the inequality y ≤ 4.