Question

Which point is a solution to the simultaneous inequalities y<1/2x and y>1/3x+3?
(1 point)

(25,12)
(25,12)

(3,6)
(3,6)

(2,2)
(2,2)

(10,5)

Answers

Answered by GPT-4o mini
To determine which point is a solution to the simultaneous inequalities \(y < \frac{1}{2}x\) and \(y > \frac{1}{3}x + 3\), we can check each point.

1. **Point (25, 12)**:
- Check \(y < \frac{1}{2}x\):
\[
12 < \frac{1}{2}(25) \quad \Rightarrow \quad 12 < 12.5 \quad \text{(True)}
\]
- Check \(y > \frac{1}{3}x + 3\):
\[
12 > \frac{1}{3}(25) + 3 \quad \Rightarrow \quad 12 > \frac{25}{3} + 3 \quad \Rightarrow \quad 12 > 8.33 \quad \text{(True)}
\]
- So (25, 12) is a solution.

2. **Point (3, 6)**:
- Check \(y < \frac{1}{2}x\):
\[
6 < \frac{1}{2}(3) \quad \Rightarrow \quad 6 < 1.5 \quad \text{(False)}
\]
- So (3, 6) is not a solution.

3. **Point (2, 2)**:
- Check \(y < \frac{1}{2}x\):
\[
2 < \frac{1}{2}(2) \quad \Rightarrow \quad 2 < 1 \quad \text{(False)}
\]
- So (2, 2) is not a solution.

4. **Point (10, 5)**:
- Check \(y < \frac{1}{2}x\):
\[
5 < \frac{1}{2}(10) \quad \Rightarrow \quad 5 < 5 \quad \text{(False)}
\]
- So (10, 5) is not a solution.

The only point that satisfies both inequalities is **(25, 12)**.
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