Solve the linear inequalities.
`2x + y ge 4, x + y le 3, 2x - 3y le 6`

Explanation

To solve the linear inequalities \(2x + y \geq 4\), \(x + y \leq 3\), and \(2x - 3y \leq 6\), we will follow these steps:

### Step 1: Convert inequalities to equations
We first convert each inequality into an equation to find the boundary lines.

1. **First inequality**: \(2x + y = 4\)
2. **Second inequality**: \(x + y = 3\)
3. **Third inequality**: \(2x - 3y = 6\)

### Step 2: Find intercepts for each equation
We will find the x-intercept and y-intercept for each equation.

1. **For \(2x + y = 4\)**:
- **x-intercept**: Set \(y = 0\):
\[ 2x + 0 = 4 \implies x = 2 \quad \text{(Point: (2, 0))} \]
- **y-intercept**: Set \(x = 0\):
\[ 2(0) + y = 4 \implies y = 4 \quad \text{(Point: (0, 4))} \]

2. **For \(x + y = 3\)**:
- **x-intercept**: Set \(y = 0\):
\[ x + 0 = 3 \implies x = 3 \quad \text{(Point: (3, 0))} \]
- **y-intercept**: Set \(x = 0\):
\[ 0 + y = 3 \implies y = 3 \quad \text{(Point: (0, 3))} \]

3. **For \(2x - 3y = 6\)**:
- **x-intercept**: Set \(y = 0\):
\[ 2x - 3(0) = 6 \implies 2x = 6 \implies x = 3 \quad \text{(Point: (3, 0))} \]
- **y-intercept**: Set \(x = 0\):
\[ 2(0) - 3y = 6 \implies -3y = 6 \implies y = -2 \quad \text{(Point: (0, -2))} \]

### Step 3: Plot the lines on a graph
Now we plot the points on a graph and draw the lines:

- For \(2x + y = 4\): Points (2, 0) and (0, 4)
- For \(x + y = 3\): Points (3, 0) and (0, 3)
- For \(2x - 3y = 6\): Points (3, 0) and (0, -2)

### Step 4: Determine the regions for each inequality
Next, we need to determine which side of each line satisfies the inequality:

1. **For \(2x + y \geq 4\)**:
- Choose a test point, e.g., (0, 0):
\[ 2(0) + 0 \geq 4 \implies 0 \geq 4 \quad \text{(False)} \]
- The solution lies on the side opposite to the origin.

2. **For \(x + y \leq 3\)**:
- Choose a test point, e.g., (0, 0):
\[ 0 + 0 \leq 3 \quad \text{(True)} \]
- The solution lies on the side including the origin.

3. **For \(2x - 3y \leq 6\)**:
- Choose a test point, e.g., (0, 0):
\[ 2(0) - 3(0) \leq 6 \quad \text{(True)} \]
- The solution lies on the side including the origin.

### Step 5: Identify the feasible region
The feasible region is where all the shaded areas from the inequalities overlap. This region will be bounded by the lines we plotted.

### Step 6: Conclusion
The solution to the system of inequalities is the region where all three inequalities are satisfied. This region can be shaded on the graph.