The point (7,8) lies in the half-plane `2x + 3y - 12 ge 0`.

Answer

To determine whether the point (7, 8) lies in the half-plane defined by the inequality \(2x + 3y - 12 \geq 0\), we will follow these steps:

### Step 1: Substitute the coordinates into the inequality
We need to substitute \(x = 7\) and \(y = 8\) into the inequality \(2x + 3y - 12\).

\[ 2(7) + 3(8) - 12 \]

### Step 2: Calculate the expression
Now, we will calculate the left-hand side of the inequality.

\[ = 14 + 24 - 12 \]

### Step 3: Simplify the expression
Next, we simplify the expression:

\[ = 14 + 24 - 12 = 26 \]

### Step 4: Check the inequality
Now, we need to check if this value satisfies the inequality \(26 \geq 0\).

### Step 5: Conclusion
Since \(26\) is indeed greater than \(0\), we conclude that the point (7, 8) satisfies the inequality \(2x + 3y - 12 \geq 0\). Therefore, the point (7, 8) lies in the half-plane defined by the inequality.

### Final Statement
Thus, the statement is true: the point (7, 8) lies in the half-plane \(2x + 3y - 12 \geq 0\).

---