Question

which of the following points shows a solution to the simultaneous inequalities y > 1/2 x + 5 and y < -2x + 1?
a (1,5.5)
b (-5,5)
c (4,8)
d (-1.6,4.2)

Answer 1

To find the points that satisfy the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will evaluate each point one by one.

1. **For point a (1, 5.5)**:
- Check \( y > \frac{1}{2}x + 5 \):
\[
5.5 > \frac{1}{2}(1) + 5 \Rightarrow 5.5 > 0.5 + 5 \Rightarrow 5.5 > 5.5 \text{ (False)}
\]
- So, point **a** does not satisfy the inequalities.

2. **For point b (-5, 5)**:
- Check \( y > \frac{1}{2}x + 5 \):
\[
5 > \frac{1}{2}(-5) + 5 \Rightarrow 5 > -2.5 + 5 \Rightarrow 5 > 2.5 \text{ (True)}
\]
- Check \( y < -2x + 1 \):
\[
5 < -2(-5) + 1 \Rightarrow 5 < 10 + 1 \Rightarrow 5 < 11 \text{ (True)}
\]
- Point **b** satisfies both inequalities.

3. **For point c (4, 8)**:
- Check \( y > \frac{1}{2}x + 5 \):
\[
8 > \frac{1}{2}(4) + 5 \Rightarrow 8 > 2 + 5 \Rightarrow 8 > 7 \text{ (True)}
\]
- Check \( y < -2x + 1 \):
\[
8 < -2(4) + 1 \Rightarrow 8 < -8 + 1 \Rightarrow 8 < -7 \text{ (False)}
\]
- Point **c** does not satisfy both inequalities.

4. **For point d (-1.6, 4.2)**:
- Check \( y > \frac{1}{2}x + 5 \):
\[
4.2 > \frac{1}{2}(-1.6) + 5 \Rightarrow 4.2 > -0.8 + 5 \Rightarrow 4.2 > 4.2 \text{ (False)}
\]
- So, point **d** does not satisfy the inequalities.

In conclusion, the only point that shows a solution to the simultaneous inequalities is:
**b (-5, 5)**.