All points lying inside the triangle formed by the point (1,3),(5,0) and (-1,2) satisfy

Options

A. `3x+2yge0`
B. `2x+y-13ge0`
C. `2x-3y-12le0`
D. `-2x+yge0`

Answer

To determine which inequalities are satisfied by all points lying inside the triangle formed by the points (1, 3), (5, 0), and (-1, 2), we will analyze each inequality one by one.

### Step 1: Identify the vertices of the triangle
The vertices of the triangle are given as:
- A(1, 3)
- B(5, 0)
- C(-1, 2)

### Step 2: Plot the points and draw the triangle
Plot the points A, B, and C on a Cartesian plane and connect them to form triangle ABC.

### Step 3: Analyze the inequalities
We will analyze each inequality to determine if the points inside the triangle satisfy it.

#### Inequality 1: \(3x + 2y \geq 0\)
1. **Find the corresponding line**: Set \(3x + 2y = 0\) to find the line.
- Rearranging gives \(y = -\frac{3}{2}x\).
- This line passes through the origin (0, 0) and has a slope of \(-\frac{3}{2}\).

2. **Test a point inside the triangle**: Let's test the point (1, 3).
- Substitute into the inequality: \(3(1) + 2(3) = 3 + 6 = 9\), which is greater than 0.
- Since (1, 3) satisfies the inequality, points inside the triangle will also satisfy it.

#### Inequality 2: \(2x + y \leq 13\)
1. **Find the corresponding line**: Set \(2x + y = 13\).
- Rearranging gives \(y = -2x + 13\).

2. **Test a point inside the triangle**: Let's test the point (1, 3).
- Substitute into the inequality: \(2(1) + 3 = 2 + 3 = 5\), which is less than 13.
- Since (1, 3) satisfies the inequality, points inside the triangle will also satisfy it.

#### Inequality 3: \(2x - 3y \geq -12\)
1. **Find the corresponding line**: Set \(2x - 3y = -12\).
- Rearranging gives \(y = \frac{2}{3}x + 4\).

2. **Test a point inside the triangle**: Let's test the point (1, 3).
- Substitute into the inequality: \(2(1) - 3(3) = 2 - 9 = -7\), which is not greater than -12.
- Since (1, 3) does not satisfy the inequality, points inside the triangle will not satisfy it.

#### Inequality 4: \(-2x + y \geq 0\)
1. **Find the corresponding line**: Set \(-2x + y = 0\).
- Rearranging gives \(y = 2x\).

2. **Test a point inside the triangle**: Let's test the point (1, 3).
- Substitute into the inequality: \(-2(1) + 3 = -2 + 3 = 1\), which is greater than 0.
- Since (1, 3) satisfies the inequality, points inside the triangle will also satisfy it.

### Conclusion
The inequalities satisfied by all points lying inside the triangle formed by the points (1, 3), (5, 0), and (-1, 2) are:
1. \(3x + 2y \geq 0\)
2. \(2x + y \leq 13\)
3. \(-2x + y \geq 0\)

### Final Answer
The inequalities that are satisfied by all points lying inside the triangle are:
1. \(3x + 2y \geq 0\)
2. \(2x + y \leq 13\)
3. \(-2x + y \geq 0\)