The point, which lies in the half-plane `2x + 3y - 12 ge 0` is :

Options

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

Answer

To determine which point lies in the half-plane defined by the inequality \(2x + 3y - 12 \geq 0\), we will follow these steps:

### Step 1: Identify the boundary line
The boundary line of the inequality can be found by setting the equation \(2x + 3y - 12 = 0\). This line divides the plane into two regions.

### Step 2: Find the intercepts of the line
To find the intercepts, we can set \(y = 0\) to find the x-intercept and \(x = 0\) to find the y-intercept.

- **X-intercept**: Set \(y = 0\)
\[ 2x + 3(0) - 12 = 0 \implies 2x = 12 \implies x = 6 \]
So, the x-intercept is \((6, 0)\).

- **Y-intercept**: Set \(x = 0\)
\[ 2(0) + 3y - 12 = 0 \implies 3y = 12 \implies y = 4 \]
So, the y-intercept is \((0, 4)\).

### Step 3: Plot the line and identify the half-plane
The line passes through the points \((6, 0)\) and \((0, 4)\). The half-plane defined by the inequality \(2x + 3y - 12 \geq 0\) is the region above this line.

### Step 4: Test the given points
Now we will test the provided points to see if they satisfy the inequality \(2x + 3y - 12 \geq 0\).

1. **Point (1, 2)**:
\[ 2(1) + 3(2) - 12 = 2 + 6 - 12 = -4 \quad (\text{Not in the half-plane}) \]

2. **Point (2, 1)**:
\[ 2(2) + 3(1) - 12 = 4 + 3 - 12 = -5 \quad (\text{Not in the half-plane}) \]

3. **Point (2, 2)**:
\[ 2(2) + 3(2) - 12 = 4 + 6 - 12 = -2 \quad (\text{Not in the half-plane}) \]

4. **Point (2, 3)**:
\[ 2(2) + 3(3) - 12 = 4 + 9 - 12 = 1 \quad (\text{In the half-plane}) \]

### Conclusion
The point that lies in the half-plane defined by the inequality \(2x + 3y - 12 \geq 0\) is \((2, 3)\).

---