Question

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

A: ( 3,6)

B:( 10,5)

C: ( 2,2)

D: (25,12)

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 points given.

### Inequalities
1. \( y < \frac{1}{2}x \)
2. \( y > \frac{1}{3}x + 3 \)

### Checking each point

**A: (3, 6)**
- For \( y < \frac{1}{2}x \):
\( 6 < \frac{1}{2}(3) \)
\( 6 < 1.5 \) (False)

- For \( y > \frac{1}{3}x + 3 \):
\( 6 > \frac{1}{3}(3) + 3 \)
\( 6 > 1 + 3 \)
\( 6 > 4 \) (True)

Result: False (doesn't satisfy both inequalities)

**B: (10, 5)**
- For \( y < \frac{1}{2}x \):
\( 5 < \frac{1}{2}(10) \)
\( 5 < 5 \) (False)

- For \( y > \frac{1}{3}x + 3 \):
\( 5 > \frac{1}{3}(10) + 3 \)
\( 5 > \frac{10}{3} + 3 \)
\( 5 > 3.33 + 3 \)
\( 5 > 6.33 \) (False)

Result: False (doesn't satisfy both inequalities)

**C: (2, 2)**
- For \( y < \frac{1}{2}x \):
\( 2 < \frac{1}{2}(2) \)
\( 2 < 1 \) (False)

- For \( y > \frac{1}{3}x + 3 \):
\( 2 > \frac{1}{3}(2) + 3 \)
\( 2 > \frac{2}{3} + 3 \)
\( 2 > 0.67 + 3 \)
\( 2 > 3.67 \) (False)

Result: False (doesn't satisfy both inequalities)

**D: (25, 12)**
- For \( y < \frac{1}{2}x \):
\( 12 < \frac{1}{2}(25) \)
\( 12 < 12.5 \) (True)

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

Result: True (satisfies both inequalities)

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