Question

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

A: ( -1.6, 4.2)

B: ( -5,5)

C: ( 4,8)

D: ( 1,5.5)

Answer 1

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)**.