Question

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

(1 point)
Responses

(25,12)
left parenthesis 25 comma 12 right parenthesis

(10,5)
left parenthesis 10 comma 5 right parenthesis

(2,2)
left parenthesis 2 comma 2 right parenthesis

(3,6)

Answer 1

To determine which point is a solution to the simultaneous inequalities \( y < \frac{1}{2}x \) and \( y > \frac{1}{3}x + 3 \), we will evaluate each of the given points.

1. **Point (25, 12)**:
- Check \( y < \frac{1}{2}x \):
\( 12 < \frac{1}{2} \times 25 \) → \( 12 < 12.5 \) (True)
- Check \( y > \frac{1}{3}x + 3 \):
\( 12 > \frac{1}{3} \times 25 + 3 \) → \( 12 > 8.33 + 3 \) → \( 12 > 11.33 \) (True)

Both inequalities are satisfied. So, **(25, 12)** is a solution.

2. **Point (10, 5)**:
- Check \( y < \frac{1}{2}x \):
\( 5 < \frac{1}{2} \times 10 \) → \( 5 < 5 \) (False)
- Check \( y > \frac{1}{3}x + 3 \):
\( 5 > \frac{1}{3} \times 10 + 3 \) → \( 5 > 3.33 + 3 \) → \( 5 > 6.33 \) (False)

This point does not satisfy either inequality.

3. **Point (2, 2)**:
- Check \( y < \frac{1}{2}x \):
\( 2 < \frac{1}{2} \times 2 \) → \( 2 < 1 \) (False)
- Check \( y > \frac{1}{3}x + 3 \):
\( 2 > \frac{1}{3} \times 2 + 3 \) → \( 2 > 0.67 + 3 \) → \( 2 > 3.67 \) (False)

This point does not satisfy either inequality.

4. **Point (3, 6)**:
- Check \( y < \frac{1}{2}x \):
\( 6 < \frac{1}{2} \times 3 \) → \( 6 < 1.5 \) (False)
- Check \( y > \frac{1}{3}x + 3 \):
\( 6 > \frac{1}{3} \times 3 + 3 \) → \( 6 > 1 + 3 \) → \( 6 > 4 \) (True)

This point satisfies only one inequality.

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