If the feasible region is bounded by the inequations `2x + 3y le 12 , 2x + y le 8, 0 le x, 0 le y` then point (5,4) is ________ of the feasible region.

Answer

To determine whether the point (5, 4) is part of the feasible region defined by the inequalities \(2x + 3y \leq 12\), \(2x + y \leq 8\), \(0 \leq x\), and \(0 \leq y\), we will follow these steps:

### Step 1: Check the inequalities for the point (5, 4)
We will substitute \(x = 5\) and \(y = 4\) into each inequality.

1. **First Inequality: \(2x + 3y \leq 12\)**
\[ 2(5) + 3(4) \leq 12 \\ 10 + 12 \leq 12 \\ 22 \leq 12 \quad \text{(False)} \]

2. **Second Inequality: \(2x + y \leq 8\)**
\[ 2(5) + 4 \leq 8 \\ 10 + 4 \leq 8 \\ 14 \leq 8 \quad \text{(False)} \]

3. **Non-negativity Constraints:**
\[ 0 \leq 5 \quad \text{(True)} \\ 0 \leq 4 \quad \text{(True)} \]

### Step 2: Analyze the results
- The point (5, 4) does not satisfy the first inequality \(2x + 3y \leq 12\) and the second inequality \(2x + y \leq 8\).
- Although it satisfies the non-negativity constraints, it fails to meet the other two inequalities.

### Conclusion
Since the point (5, 4) does not satisfy all the inequalities that define the feasible region, we conclude that the point (5, 4) is **not a part of the feasible region**.

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