Which of the following points shows a solution to the simultaneous inequalities y > 1/2x + 5 and y < -2x + 1?

Answer 1

To determine which of the given points satisfy both inequalities, we can substitute the values of the points into the inequalities and check if the resulting statements are true.

Let's test each point:

Point A(0, 4):
For the first inequality:
y > 1/2x + 5
Substituting the values of point A:
4 > 1/2(0) + 5
4 > 0 + 5
4 > 5
This statement is false.

For the second inequality:
y < -2x + 1
Substituting the values of point A:
4 < -2(0) + 1
4 < 0 + 1
4 < 1
This statement is also false.

Since neither inequality is true for Point A, it is not a solution.

Point B(-1, -4):
For the first inequality:
y > 1/2x + 5
Substituting the values of point B:
-4 > 1/2(-1) + 5
-4 > -1/2 + 5
-4 > 4.5
This statement is false.

For the second inequality:
y < -2x + 1
Substituting the values of point B:
-4 < -2(-1) + 1
-4 < 2 + 1
-4 < 3
This statement is true.

Since the second inequality is true for Point B, it is not a solution.

Point C(2, -5):
For the first inequality:
y > 1/2x + 5
Substituting the values of point C:
-5 > 1/2(2) + 5
-5 > 1 + 5
-5 > 6
This statement is false.

For the second inequality:
y < -2x + 1
Substituting the values of point C:
-5 < -2(2) + 1
-5 < -4 + 1
-5 < -3
This statement is true.

Since the second inequality is true for Point C, it is not a solution.

Point D(6, -12):
For the first inequality:
y > 1/2x + 5
Substituting the values of point D:
-12 > 1/2(6) + 5
-12 > 3 + 5
-12 > 8
This statement is false.

For the second inequality:
y < -2x + 1
Substituting the values of point D:
-12 < -2(6) + 1
-12 < -12 + 1
-12 < -11
This statement is true.

Since the second inequality is true for Point D, it is not a solution.

Based on the analysis, Point B(-1, -4) is the only point that satisfies both inequalities: y > 1/2x + 5 and y < -2x + 1.