Which point is a solution to this system of inequalities?
y≤1/2x-3
y+2x>6
A.
(7,-8)
B.
(2,-3)
C.
(5,-2)
D.
(4,1)
To determine which point is a solution to the system of inequalities, we can substitute the coordinates of each point into the inequalities and see if the resulting statements are true or false.
1. Point (7, -8):
Substituting the coordinates into the first inequality:
-8 ≤ (1/2)(7) - 3
-8 ≤ 3.5 - 3
-8 ≤ 0.5 (True)
Substituting the coordinates into the second inequality:
-8 + 2(7) > 6
-8 + 14 > 6
6 > 6 (False)
2. Point (2, -3):
Substituting the coordinates into the first inequality:
-3 ≤ (1/2)(2) - 3
-3 ≤ 1 - 3
-3 ≤ -2 (True)
Substituting the coordinates into the second inequality:
-3 + 2(2) > 6
-3 + 4 > 6
1 > 6 (False)
3. Point (5, -2):
Substituting the coordinates into the first inequality:
-2 ≤ (1/2)(5) - 3
-2 ≤ 2.5 - 3
-2 ≤ -0.5 (True)
Substituting the coordinates into the second inequality:
-2 + 2(5) > 6
-2 + 10 > 6
8 > 6 (True)
4. Point (4, 1):
Substituting the coordinates into the first inequality:
1 ≤ (1/2)(4) - 3
1 ≤ 2 - 3
1 ≤ -1 (False)
Substituting the coordinates into the second inequality:
1 + 2(4) > 6
1 + 8 > 6
9 > 6 (True)
Based on these calculations, the point (5, -2) is the only solution to the system of inequalities. Therefore, the correct answer is C. (5, -2).