`x + y le 6, x + y ge 4`

Explanation

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

### Step 1: Write down the inequalities
The inequalities given are:
1. \( x + y \leq 6 \)
2. \( x + y \geq 4 \)

### Step 2: Convert inequalities to equations
To plot the graphs, we first convert the inequalities to equations:
1. \( x + y = 6 \)
2. \( x + y = 4 \)

### Step 3: Find points for the first equation \( x + y = 6 \)
To plot the line \( x + y = 6 \), we can find two points:
- Let \( x = 0 \):
\[ 0 + y = 6 \Rightarrow y = 6 \quad \text{(Point: (0, 6))} \]
- Let \( y = 0 \):
\[ x + 0 = 6 \Rightarrow x = 6 \quad \text{(Point: (6, 0))} \]

### Step 4: Plot the first line
Plot the points (0, 6) and (6, 0) on the coordinate plane and draw a solid line through these points since the inequality is less than or equal to (≤).

### Step 5: Find points for the second equation \( x + y = 4 \)
Next, we find two points for the line \( x + y = 4 \):
- Let \( x = 0 \):
\[ 0 + y = 4 \Rightarrow y = 4 \quad \text{(Point: (0, 4))} \]
- Let \( y = 0 \):
\[ x + 0 = 4 \Rightarrow x = 4 \quad \text{(Point: (4, 0))} \]

### Step 6: Plot the second line
Plot the points (0, 4) and (4, 0) on the coordinate plane and draw a solid line through these points since the inequality is greater than or equal to (≥).

### Step 7: Determine the solution regions
Now we need to determine the regions defined by the inequalities:
- For \( x + y \leq 6 \), the region is below the line \( x + y = 6 \).
- For \( x + y \geq 4 \), the region is above the line \( x + y = 4 \).

### Step 8: Identify the common region
The solution to the system of inequalities is the region where the two shaded areas overlap. This region is bounded by the lines \( x + y = 6 \) and \( x + y = 4 \).

### Step 9: Conclusion
The solution region is the area between the two lines, including the lines themselves since the inequalities are non-strict (≤ and ≥).

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