All of the points lying inside thr triangle formed by the points (0,4) (2,5) and (6,2) satisfy

Options

A. `3x +2y+8ge0`
B. `2x+y - 10 ge 0`
C. `2x-3y -11 ge0`
D. ` -2x + y - 3 ge 0`

Answer

To determine which of the given inequalities is satisfied by all points lying inside the triangle formed by the points (0, 4), (2, 5), and (6, 2), we will evaluate each option step by step.

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- A(0, 4)
- B(2, 5)
- C(6, 2)

### Step 2: Evaluate Option 1: \(3x + 2y + 8 \geq 0\)
We will check if this inequality holds for all three vertices.

1. For point A(0, 4):
\[ 3(0) + 2(4) + 8 = 0 + 8 + 8 = 16 \quad (\text{which is } \geq 0) \]

2. For point B(2, 5):
\[ 3(2) + 2(5) + 8 = 6 + 10 + 8 = 24 \quad (\text{which is } \geq 0) \]

3. For point C(6, 2):
\[ 3(6) + 2(2) + 8 = 18 + 4 + 8 = 30 \quad (\text{which is } \geq 0) \]

Since all points satisfy the inequality \(3x + 2y + 8 \geq 0\), this option is valid.

### Step 3: Evaluate Option 2: \(2x + y - 10 \geq 0\)
Now, we check this inequality for the vertices.

1. For point A(0, 4):
\[ 2(0) + 4 - 10 = 0 + 4 - 10 = -6 \quad (\text{which is } < 0) \]

Since point A does not satisfy the inequality, we can conclude that this option is false.

### Step 4: Evaluate Option 3: \(2x - 3y - 11 \geq 0\)
Next, we check this inequality.

1. For point A(0, 4):
\[ 2(0) - 3(4) - 11 = 0 - 12 - 11 = -23 \quad (\text{which is } < 0) \]

Since point A does not satisfy the inequality, this option is also false.

### Step 5: Evaluate Option 4: \(-2x + y - 3 \geq 0\)
Finally, we check this inequality.

1. For point A(0, 4):
\[ -2(0) + 4 - 3 = 0 + 4 - 3 = 1 \quad (\text{which is } \geq 0) \]

2. For point B(2, 5):
\[ -2(2) + 5 - 3 = -4 + 5 - 3 = -2 \quad (\text{which is } < 0) \]

Since point B does not satisfy the inequality, this option is false.

### Conclusion
After evaluating all options, we find that only **Option 1: \(3x + 2y + 8 \geq 0\)** is satisfied by all points inside the triangle formed by the points (0, 4), (2, 5), and (6, 2).