The half plane represented by 3x + 4y ≥ 12 includes the point (4,3). (True/False)

Answer

To determine if the point (4, 3) is included in the half-plane represented by the inequality \(3x + 4y \geq 12\), we will substitute the coordinates of the point into the inequality and check if it holds true.

### Step-by-Step Solution:

1. **Identify the Inequality**:
The inequality given is \(3x + 4y \geq 12\).

2. **Substitute the Point**:
We need to check if the point (4, 3) satisfies this inequality. Here, \(x = 4\) and \(y = 3\).
Substitute \(x\) and \(y\) into the inequality:
\[ 3(4) + 4(3) \geq 12 \]

3. **Calculate the Left Side**:
Now calculate the left-hand side:
\[ 3(4) = 12 \quad \text{and} \quad 4(3) = 12 \]
Therefore,
\[ 12 + 12 = 24 \]

4. **Compare with the Right Side**:
Now we compare the left-hand side with the right-hand side:
\[ 24 \geq 12 \]
This statement is true.

5. **Conclusion**:
Since the point (4, 3) satisfies the inequality \(3x + 4y \geq 12\), we conclude that the half-plane represented by this inequality does indeed include the point (4, 3).

Thus, the statement is **True**.