Question

Which of the following points shows a solution to the simultaneous inequalities: y > 1/2 x + 5 and < -2x + 1
A. (4, 8)
B. (-5, 5)
C. (1, 5.5)
D. (-1.6, 4.2)

Answer 1

To determine which of the given points satisfies the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will substitute each point into both inequalities.

1. **Inequality 1: \( y > \frac{1}{2}x + 5 \)**
2. **Inequality 2: \( y < -2x + 1 \)**

Let's evaluate each point:

### A. (4, 8)
- For \( y > \frac{1}{2}x + 5 \):
\( 8 > \frac{1}{2}(4) + 5 \)
\( 8 > 2 + 5 \)
\( 8 > 7 \) (True)

- For \( y < -2x + 1 \):
\( 8 < -2(4) + 1 \)
\( 8 < -8 + 1 \)
\( 8 < -7 \) (False)

### B. (-5, 5)
- For \( y > \frac{1}{2}x + 5 \):
\( 5 > \frac{1}{2}(-5) + 5 \)
\( 5 > -2.5 + 5 \)
\( 5 > 2.5 \) (True)

- For \( y < -2x + 1 \):
\( 5 < -2(-5) + 1 \)
\( 5 < 10 + 1 \)
\( 5 < 11 \) (True)

### C. (1, 5.5)
- For \( y > \frac{1}{2}x + 5 \):
\( 5.5 > \frac{1}{2}(1) + 5 \)
\( 5.5 > 0.5 + 5 \)
\( 5.5 > 5.5 \) (False)

### D. (-1.6, 4.2)
- For \( y > \frac{1}{2}x + 5 \):
\( 4.2 > \frac{1}{2}(-1.6) + 5 \)
\( 4.2 > -0.8 + 5 \)
\( 4.2 > 4.2 \) (False)

Now we can summarize the results:
- A. (4, 8): False for inequality 2
- B. (-5, 5): True for both inequalities
- C. (1, 5.5): False for inequality 1
- D. (-1.6, 4.2): False for inequality 1

The only point that satisfies both inequalities is:

**B. (-5, 5)**