which of the following points shows a solution to the simultaneous inequalities y>1/2x+5 and y<-2x+1?
pick one
a. (-5,5)
b. (-1.6,4.2)
c. (4,8)
d. (1,5.5)
To determine if a point is a solution to a system of inequalities, we substitute the coordinates of the point into each inequality and check if both inequalities are true.
a. (-5,5):
For y > 1/2x + 5:
5 > 1/2(-5) + 5
5 > -5/2 + 5
5 > -2.5 + 5
5 > 2.5 (True)
For y < -2x + 1:
5 < -2(-5) + 1
5 < 10 + 1
5 < 11 (True)
Therefore, (-5,5) satisfies both inequalities.
b. (-1.6,4.2):
For y > 1/2x + 5:
4.2 > 1/2(-1.6) + 5
4.2 > -0.8 + 5
4.2 > 4.2 (False)
For y < -2x + 1:
4.2 < -2(-1.6) + 1
4.2 < 3.2 + 1
4.2 < 4.2 (False)
Therefore, (-1.6,4.2) does not satisfy both inequalities.
c. (4,8):
For y > 1/2x + 5:
8 > 1/2(4) + 5
8 > 2 + 5
8 > 7 (True)
For y < -2x + 1:
8 < -2(4) + 1
8 < -8 + 1
8 < -7 (False)
Therefore, (4,8) does not satisfy both inequalities.
d. (1,5.5):
For y > 1/2x + 5:
5.5 > 1/2(1) + 5
5.5 > 0.5 + 5
5.5 > 5.5 (False)
For y < -2x + 1:
5.5 < -2(1) + 1
5.5 < -2 + 1
5.5 < -1 (False)
Therefore, (1,5.5) does not satisfy both inequalities.
The only point that satisfies both inequalities is (-5,5).
Therefore, the correct answer is option a.