The point, which does not lie in the half-plane `3x + 4y - 15 le 0` is:

Options

A. (1, 2)
B. (2, 1)
C. (2, 2)
D. (3, 2)

Answer

To determine which point does not lie in the half-plane defined by the inequality \(3x + 4y - 15 \leq 0\), we will evaluate several points by substituting their coordinates into the inequality.

### Step-by-Step Solution:

1. **Understand the Inequality**: The inequality \(3x + 4y - 15 \leq 0\) defines a half-plane. Points that satisfy this inequality will lie in or on the boundary of the half-plane.

2. **Rearranging the Inequality**: We can rearrange the inequality to express it in a more familiar form:
\[ 3x + 4y \leq 15 \]

3. **Choose Points to Test**: We will test the following points:
- Point A: (1, 2)
- Point B: (2, 1)
- Point C: (2, 2)
- Point D: (3, 2)

4. **Test Point A (1, 2)**:
- Substitute \(x = 1\) and \(y = 2\):
\[ 3(1) + 4(2) = 3 + 8 = 11 \]
- Check the inequality:
\[ 11 \leq 15 \quad \text{(True)} \]
- Point A lies in the half-plane.

5. **Test Point B (2, 1)**:
- Substitute \(x = 2\) and \(y = 1\):
\[ 3(2) + 4(1) = 6 + 4 = 10 \]
- Check the inequality:
\[ 10 \leq 15 \quad \text{(True)} \]
- Point B lies in the half-plane.

6. **Test Point C (2, 2)**:
- Substitute \(x = 2\) and \(y = 2\):
\[ 3(2) + 4(2) = 6 + 8 = 14 \]
- Check the inequality:
\[ 14 \leq 15 \quad \text{(True)} \]
- Point C lies in the half-plane.

7. **Test Point D (3, 2)**:
- Substitute \(x = 3\) and \(y = 2\):
\[ 3(3) + 4(2) = 9 + 8 = 17 \]
- Check the inequality:
\[ 17 \leq 15 \quad \text{(False)} \]
- Point D does not lie in the half-plane.

8. **Conclusion**: The point that does not lie in the half-plane defined by \(3x + 4y - 15 \leq 0\) is \((3, 2)\).