The points (1,1/2) and (3,-1/2) are

Given:

Points: \( A = (1, \frac{1}{2}) \), \( B = (3, -\frac{1}{2}) \)

Line: \( 2x + 3y = k \)

Step 1: Evaluate \( 2x + 3y \)

For A: \( 2(1) + 3\left(\frac{1}{2}\right) = \frac{7}{2} \)
For B: \( 2(3) + 3\left(-\frac{1}{2}\right) = \frac{9}{2} \)

✅ Option-wise Check:

• In between the lines \( 2x + 3y = -6 \) and \( 2x + 3y = 6 \): ✔️ True since \( \frac{7}{2}, \frac{9}{2} \in (-6, 6) \)
• On the same side of \( 2x + 3y = 6 \): ✔️ True, both values are less than 6
• On the same side of \( 2x + 3y = -6 \): ✔️ True, both values are greater than -6
• On the opposite side of \( 2x + 3y = -6 \): ❌ False, both are on the same side

✅ Final Answer:

The correct statements are:

1. In between the lines \( 2x + 3y = -6 \) and \( 2x + 3y = 6 \)
2. On the same side of the line \( 2x + 3y = 6 \)
3. On the same side of the line \( 2x + 3y = -6 \)