Question

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

(1 point)
Responses

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

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

(3,6)
left parenthesis 3 comma 6 right parenthesis

(10,5)

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 substitute each point into the inequalities and check if both conditions are satisfied.

1. **Point (2, 2)**:
- For \( y < \frac{1}{2}x \):
- \( 2 < \frac{1}{2}(2) \) ⇒ \( 2 < 1 \) (False)
- This point does not satisfy the first inequality.

2. **Point (25, 12)**:
- For \( y < \frac{1}{2}x \):
- \( 12 < \frac{1}{2}(25) \) ⇒ \( 12 < 12.5 \) (True)
- For \( y > \frac{1}{3}x + 3 \):
- \( 12 > \frac{1}{3}(25) + 3 \)
- \( 12 > \frac{25}{3} + 3 \) ⇒ \( 12 > \frac{25}{3} + \frac{9}{3} \)
- \( 12 > \frac{34}{3} \) ⇒ \( 12 > 11.33 \) (True)
- This point satisfies both inequalities.

3. **Point (3, 6)**:
- For \( y < \frac{1}{2}x \):
- \( 6 < \frac{1}{2}(3) \) ⇒ \( 6 < 1.5 \) (False)
- This point does not satisfy the first inequality.

4. **Point (10, 5)**:
- For \( y < \frac{1}{2}x \):
- \( 5 < \frac{1}{2}(10) \) ⇒ \( 5 < 5 \) (False)
- This point does not satisfy the first inequality.

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