Question

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

(−1.6,4.2)
left parenthesis negative 1.6 comma 4.2 right parenthesis

(−5,5)
left parenthesis negative 5 comma 5 right parenthesis

(4,8)
left parenthesis 4 comma 8 right parenthesis

(1,5.5)

Answer 1

To determine which points satisfy the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we can evaluate each option:

1. **For the point \((-1.6, 4.2)\)**:
- Check \(y > \frac{1}{2}x + 5\):
\[
4.2 > \frac{1}{2}(-1.6) + 5 \\
4.2 > -0.8 + 5 \\
4.2 > 4.2 \quad \text{(false)}
\]
- This point does not satisfy the first inequality.

2. **For the point \((-5, 5)\)**:
- Check \(y > \frac{1}{2}x + 5\):
\[
5 > \frac{1}{2}(-5) + 5 \\
5 > -2.5 + 5 \\
5 > 2.5 \quad \text{(true)}
\]
- Check \(y < -2x + 1\):
\[
5 < -2(-5) + 1 \\
5 < 10 + 1 \\
5 < 11 \quad \text{(true)}
\]
- This point satisfies both inequalities.

3. **For the point \((4, 8)\)**:
- Check \(y > \frac{1}{2}x + 5\):
\[
8 > \frac{1}{2}(4) + 5 \\
8 > 2 + 5 \\
8 > 7 \quad \text{(true)}
\]
- Check \(y < -2x + 1\):
\[
8 < -2(4) + 1 \\
8 < -8 + 1 \\
8 < -7 \quad \text{(false)}
\]
- This point does not satisfy the second inequality.

4. **For the point \((1, 5.5)\)**:
- Check \(y > \frac{1}{2}x + 5\):
\[
5.5 > \frac{1}{2}(1) + 5 \\
5.5 > 0.5 + 5 \\
5.5 > 5.5 \quad \text{(false)}
\]
- This point does not satisfy the first inequality.

The point that satisfies both inequalities is \((-5, 5)\).