Question

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

(1 point)

(25,12)

(10.5)

(3.6)

(2.2)

Answer 1

To determine which point is a solution to the inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we can substitute each point into the inequalities.

### Inequality 1: \( y < \frac{1}{2}x \)

### Inequality 2: \( y > \frac{1}{3}x + 3 \)

Let's check each point one by one.

1. **Point (25, 12)**

- \( y < \frac{1}{2}x \)
- \( 12 < \frac{1}{2}(25) \)
- \( 12 < 12.5 \) → True

- \( y > \frac{1}{3}x + 3 \)
- \( 12 > \frac{1}{3}(25) + 3 \)
- \( 12 > 8.33 + 3 \)
- \( 12 > 11.33 \) → True

→ **(25, 12)** is a solution.

2. **Point (10, 5)**

- \( y < \frac{1}{2}x \)
- \( 5 < \frac{1}{2}(10) \)
- \( 5 < 5 \) → False

- Since the first inequality is false, we can stop checking this point.

3. **Point (3, 6)**

- \( y < \frac{1}{2}x \)
- \( 6 < \frac{1}{2}(3) \)
- \( 6 < 1.5 \) → False

- Since the first inequality is false, we can stop checking this point.

4. **Point (2, 2)**

- \( y < \frac{1}{2}x \)
- \( 2 < \frac{1}{2}(2) \)
- \( 2 < 1 \) → False

- Since the first inequality is false, we can stop checking this point.

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