Select all of the ordered pairs that are solutions to the following system of inequalities.
8y ≥ – 2x – 32
4x – 8y > 0
a. (0, 2)
b. (-2, -2)
c. (2, 0)
d. (4, -5)
The solutions are the pairs of x and y values that satisfy both inequalities when substituted into the equations.
Let's start with the first inequality:
8y ≥ -2x - 32
Dividing by 8 gives us:
y ≥ -1/4x - 4
Now, let's see which of the given ordered pairs satisfy this inequality:
a. (0, 2): 2 ≥ -1/4(0) - 4 -> 2 ≥ -4 (True)
b. (-2, -2): -2 ≥ -1/4(-2) - 4 -> -2 ≥ -0.5 - 4 -> -2 ≥ -4.5 (False)
c. (2, 0): 0 ≥ -1/4(2) - 4 -> 0 ≥ -0.5 - 4 -> 0 ≥ -4.5 (True)
d. (4, -5): -5 ≥ -1/4(4) - 4 -> -5 ≥ -1 - 4 -> -5 ≥ -5 (True)
Now let's check the second inequality:
4x - 8y > 0
Dividing by 4 gives us:
x - 2y > 0
Checking the ordered pairs for this inequality:
a. (0, 2): 0 - 2(2) > 0 -> 0 - 4 > 0 -> -4 > 0 (False)
b. (-2, -2): -2 - 2(-2) > 0 -> -2 + 4 > 0 -> 2 > 0 (True)
c. (2, 0): 2 - 2(0) > 0 -> 2 - 0 > 0 -> 2 > 0 (True)
d. (4, -5): 4 - 2(-5) > 0 -> 4 + 10 > 0 -> 14 > 0 (True)
Only (-2, -2), (2, 0), and (4, -5) satisfy both inequalities.