Find common region of given inequation `x + y le 9, x gt y, xge 0 `

Explanation

To find the common region of the given inequalities \( x + y \leq 9 \), \( x > y \), and \( x \geq 0 \), we will follow these steps:

### Step 1: Graph the inequality \( x + y \leq 9 \)

1. **Convert the inequality to an equation**: The corresponding equation is \( x + y = 9 \).
2. **Find the intercepts**:
- When \( x = 0 \), \( y = 9 \) (point \( (0, 9) \)).
- When \( y = 0 \), \( x = 9 \) (point \( (9, 0) \)).
3. **Draw the line**: Plot the points \( (0, 9) \) and \( (9, 0) \) and draw a straight line through them.
4. **Shade the region**: Since the inequality is \( \leq \), shade the area below the line, including the line itself.

### Step 2: Graph the inequality \( x > y \)

1. **Convert the inequality to an equation**: The corresponding equation is \( x = y \).
2. **Draw the line**: This is a diagonal line through the origin at a 45-degree angle.
3. **Shade the region**: Since the inequality is \( > \), shade the area above the line \( x = y \) (excluding the line itself).

### Step 3: Graph the inequality \( x \geq 0 \)

1. **Convert the inequality to an equation**: The corresponding equation is \( x = 0 \) (the y-axis).
2. **Draw the line**: This is the y-axis itself.
3. **Shade the region**: Since the inequality is \( \geq \), shade the area to the right of the y-axis, including the line.

### Step 4: Identify the common region

1. **Combine the shaded areas**: The common region is where all three shaded areas overlap.
2. **Check the boundaries**: The region is bounded by:
- The line \( x + y = 9 \) (including the line),
- The area above the line \( x = y \) (excluding the line),
- The area to the right of the y-axis (including the line).

### Step 5: Describe the common region

The common region is the area that satisfies all three inequalities. It is the area below the line \( x + y = 9 \), above the line \( x = y \), and to the right of the y-axis.