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 each of the given points and check if they satisfy the inequality. The points are likely to be given as options, but since they are not specified in the question, we'll assume some common points based on the video transcript.

**Step 1: Understand the inequality**
The inequality \(3x + 4y - 15 \leq 0\) represents a half-plane in the coordinate system. We need to find points that do not satisfy this inequality.

**Step 2: Evaluate each point**

1. **Point (1, 2)**:
\[ 3(1) + 4(2) - 15 = 3 + 8 - 15 = -4 \]
Since \(-4 \leq 0\), this point satisfies the inequality.

2. **Point (2, 1)**:
\[ 3(2) + 4(1) - 15 = 6 + 4 - 15 = -5 \]
Since \(-5 \leq 0\), this point also satisfies the inequality.

3. **Point (2, 2)**:
\[ 3(2) + 4(2) - 15 = 6 + 8 - 15 = -1 \]
Since \(-1 \leq 0\), this point also satisfies the inequality.

4. **Point (3, 2)**:
\[ 3(3) + 4(2) - 15 = 9 + 8 - 15 = 2 \]
Since \(2 \leq 0\) is false, this point does not satisfy the inequality.

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