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

Answer 1

To find a solution to the simultaneous inequalities, we need to find a point that satisfies both inequalities. Let's examine the options:

a) (0, 6)
For this point, when we substitute the values into the inequalities:
- y > 1/2x + 5: 6 > 1/2(0) + 5 ⟹ 6 > 0 + 5 ⟹ 6 > 5 (true)
- y < -2x + 1: 6 < -2(0) + 1 ⟹ 6 < 0 + 1 ⟹ 6 < 1 (false)

b) (1, 3)
For this point, when we substitute the values into the inequalities:
- y > 1/2x + 5: 3 > 1/2(1) + 5 ⟹ 3 > 1/2 + 5 ⟹ 3 > 5.5 (false)
- y < -2x + 1: 3 < -2(1) + 1 ⟹ 3 < -2 + 1 ⟹ 3 < -1 (false)

c) (-2, 9)
For this point, when we substitute the values into the inequalities:
- y > 1/2x + 5: 9 > 1/2(-2) + 5 ⟹ 9 > -1 + 5 ⟹ 9 > 4 (true)
- y < -2x + 1: 9 < -2(-2) + 1 ⟹ 9 < 4 + 1 ⟹ 9 < 5 (false)

d) (-1, 4)
For this point, when we substitute the values into the inequalities:
- y > 1/2x + 5: 4 > 1/2(-1) + 5 ⟹ 4 > -0.5 + 5 ⟹ 4 > 4.5 (false)
- y < -2x + 1: 4 < -2(-1) + 1 ⟹ 4 < 2 + 1 ⟹ 4 < 3 (false)

The only point that satisfies both inequalities is the point (-2, 9).