To find which of the given points satisfies the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will check each point one by one.
1. Point A: (-1.6, 4.2)
- For \( y > \frac{1}{2}x + 5 \): \[ 4.2 > \frac{1}{2}(-1.6) + 5 \implies 4.2 > -0.8 + 5 \implies 4.2 > 4.2 \quad \text{(not satisfied)} \]
- For \( y < -2x + 1 \): \[ 4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(not satisfied)} \]
- Conclusion: Point A is not a solution.
2. Point B: (-5, 5)
- For \( y > \frac{1}{2}x + 5 \): \[ 5 > \frac{1}{2}(-5) + 5 \implies 5 > -2.5 + 5 \implies 5 > 2.5 \quad \text{(satisfied)} \]
- For \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad \text{(satisfied)} \]
- Conclusion: Point B is a solution.
3. Point C: (4, 8)
- For \( y > \frac{1}{2}x + 5 \): \[ 8 > \frac{1}{2}(4) + 5 \implies 8 > 2 + 5 \implies 8 > 7 \quad \text{(satisfied)} \]
- For \( y < -2x + 1 \): \[ 8 < -2(4) + 1 \implies 8 < -8 + 1 \implies 8 < -7 \quad \text{(not satisfied)} \]
- Conclusion: Point C is not a solution.
4. Point D: (1, 5.5)
- For \( y > \frac{1}{2}x + 5 \): \[ 5.5 > \frac{1}{2}(1) + 5 \implies 5.5 > 0.5 + 5 \implies 5.5 > 5.5 \quad \text{(not satisfied)} \]
- For \( y < -2x + 1 \): \[ 5.5 < -2(1) + 1 \implies 5.5 < -2 + 1 \implies 5.5 < -1 \quad \text{(not satisfied)} \]
- Conclusion: Point D is not a solution.
Final Conclusion: The only point that satisfies both inequalities is point B: (-5, 5).