All the points lying inside the triangle formed by the points (1, 3), (5, 6), and (-1, 2) satisfy :

Options

A. `3x+2y ge 0`
B. `2x+y+1 ge 0`
C. `-2x+11 ge 0`
D. `2x+3y-12 ge 0`

Answer

To solve the problem of determining which inequalities are satisfied by all points lying inside the triangle formed by the points \( A(1, 3) \), \( B(5, 6) \), and \( C(-1, 2) \), we will evaluate each inequality provided in the options.

### Step 1: Evaluate the first inequality \( 3x + 2y > 0 \)

1. **For point A (1, 3)**:
\[ 3(1) + 2(3) = 3 + 6 = 9 > 0 \quad \text{(True)} \]

2. **For point B (5, 6)**:
\[ 3(5) + 2(6) = 15 + 12 = 27 > 0 \quad \text{(True)} \]

3. **For point C (-1, 2)**:
\[ 3(-1) + 2(2) = -3 + 4 = 1 > 0 \quad \text{(True)} \]

Since all three points satisfy the inequality, the first inequality is valid.

### Step 2: Evaluate the second inequality \( 2x + y + 1 > 0 \)

1. **For point A (1, 3)**:
\[ 2(1) + 3 + 1 = 2 + 3 + 1 = 6 > 0 \quad \text{(True)} \]

2. **For point B (5, 6)**:
\[ 2(5) + 6 + 1 = 10 + 6 + 1 = 17 > 0 \quad \text{(True)} \]

3. **For point C (-1, 2)**:
\[ 2(-1) + 2 + 1 = -2 + 2 + 1 = 1 > 0 \quad \text{(True)} \]

Since all three points satisfy the inequality, the second inequality is also valid.

### Step 3: Evaluate the third inequality \( -2x + 11 > 0 \)

1. **For point A (1, 3)**:
\[ -2(1) + 11 = -2 + 11 = 9 > 0 \quad \text{(True)} \]

2. **For point B (5, 6)**:
\[ -2(5) + 11 = -10 + 11 = 1 > 0 \quad \text{(True)} \]

3. **For point C (-1, 2)**:
\[ -2(-1) + 11 = 2 + 11 = 13 > 0 \quad \text{(True)} \]

Since all three points satisfy the inequality, the third inequality is valid.

### Step 4: Evaluate the fourth inequality \( 2x + 3y - 12 > 0 \)

1. **For point A (1, 3)**:
\[ 2(1) + 3(3) - 12 = 2 + 9 - 12 = -1 > 0 \quad \text{(False)} \]

2. **For point B (5, 6)**:
\[ 2(5) + 3(6) - 12 = 10 + 18 - 12 = 16 > 0 \quad \text{(True)} \]

3. **For point C (-1, 2)**:
\[ 2(-1) + 3(2) - 12 = -2 + 6 - 12 = -8 > 0 \quad \text{(False)} \]

Since point A and point C do not satisfy the inequality, the fourth inequality is not valid.

### Conclusion:
The inequalities that satisfy all points inside the triangle are:
1. \( 3x + 2y > 0 \)
2. \( 2x + y + 1 > 0 \)
3. \( -2x + 11 > 0 \)

The fourth inequality \( 2x + 3y - 12 > 0 \) does not satisfy all points.