`x-y le 2, x + y le 4 , x ge 0, y ge 0 `

Explanation

To solve the given system of linear inequalities graphically, we will follow these steps:

### Step 1: Write the inequalities
We have the following inequalities:
1. \( x - y \leq 2 \)
2. \( x + y \leq 4 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert inequalities to equations
To graph the inequalities, we first convert them into equations:
1. \( x - y = 2 \)
2. \( x + y = 4 \)

### Step 3: Find intercepts for each equation
**For the first equation \( x - y = 2 \):**
- When \( x = 0 \):
\[ 0 - y = 2 \implies y = -2 \quad (\text{not in the first quadrant}) \]
- When \( y = 0 \):
\[ x - 0 = 2 \implies x = 2 \quad \text{(point (2, 0))} \]

**For the second equation \( x + y = 4 \):**
- When \( x = 0 \):
\[ 0 + y = 4 \implies y = 4 \quad \text{(point (0, 4))} \]
- When \( y = 0 \):
\[ x + 0 = 4 \implies x = 4 \quad \text{(point (4, 0))} \]

### Step 4: Plot the lines on a graph
- Plot the points (2, 0) and (0, -2) for the line \( x - y = 2 \). However, since (0, -2) is not in the first quadrant, we only consider (2, 0).
- Plot the points (0, 4) and (4, 0) for the line \( x + y = 4 \).

### Step 5: Draw the lines
- Draw a line through the points (2, 0) and (0, 4) for \( x + y = 4 \).
- Draw a line through the points (2, 0) and (4, 0) for \( x - y = 2 \).

### Step 6: Determine the shaded regions
- For \( x - y \leq 2 \): Test the point (0, 0):
\[ 0 - 0 \leq 2 \quad \text{(True)} \]
Thus, shade the region below the line \( x - y = 2 \).

- For \( x + y \leq 4 \): Test the point (0, 0):
\[ 0 + 0 \leq 4 \quad \text{(True)} \]
Thus, shade the region below the line \( x + y = 4 \).

### Step 7: Identify the feasible region
The feasible region is where the shaded areas of both inequalities overlap in the first quadrant. This region is bounded by the lines and the axes.

### Final Step: Conclusion
The solution to the system of inequalities is the area of intersection in the first quadrant, which is the required answer.

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