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You are asked to graph the feasible region for a system of inequalities:

\[
\begin{cases}
x + y \leq 9 \\
-x + y \leq 1 \\
y \geq 2
\end{cases}
\]

### Step-by-step solution:

1. **First inequality**: \( x + y \leq 9 \)

   - Rewrite as \( y \leq -x + 9 \).
   - This is a line with a slope of -1 and y-intercept 9. The region below or on the line is feasible.

2. **Second inequality**: \( -x + y \leq 1 \)

   - Rewrite as \( y \leq x + 1 \).
   - This is a line with a slope of 1 and y-intercept 1. The region below or on this line is feasible.

3. **Third inequality**: \( y \geq 2 \)

   - This is a horizontal line at \( y = 2 \). The region above or on this line is feasible.

### Plotting the lines:

- **Line 1**: \( y = -x + 9 \) 
  - Intercepts: \( x = 9 \) when \( y = 0 \), and \( y = 9 \) when \( x = 0 \).
  
- **Line 2**: \( y = x + 1 \)
  - Intercepts: \( x = -1 \) when \( y = 0 \), and \( y = 1 \) when \( x = 0 \).
  
- **Line 3**: \( y = 2 \)
  - Horizontal line at \( y = 2 \).

### Finding Intersection Points:

To find where the lines intersect:
- Intersection of \( y = -x + 9 \) and \( y = x + 1 \):
  Set \( -x + 9 = x + 1 \):
  \[
  -x + 9 = x + 1 \implies 8 = 2x \implies x = 4
  \]
  Substitute \( x = 4 \) into \( y = x + 1 \):
  \[
  y = 4 + 1 = 5
  \]
  So, the intersection is \( (4, 5) \).

- Intersection of \( y = -x + 9 \) and \( y = 2 \):
  Set \( -x + 9 = 2 \):
  \[
  -x + 9 = 2 \implies x = 7
  \]
  So, the intersection is \( (7, 2) \).

- Intersection of \( y = x + 1 \) and \( y = 2 \):
  Set \( x + 1 = 2 \):
  \[
  x = 1
  \]
  So, the intersection is \( (1, 2) \).

### Polygon vertices:

The feasible region is a polygon with vertices at:
- \( (1, 2) \)
- \( (4, 5) \)
- \( (7, 2) \)

### Graphing:

1. Draw the line \( y = -x + 9 \) with points at \( (0, 9) \) and \( (9, 0) \).
2. Draw the line \( y = x + 1 \) with points at \( (0, 1) \) and \( (-1, 0) \).
3. Draw the horizontal line \( y = 2 \).
4. Mark the polygon with vertices \( (1, 2), (4, 5), (7, 2) \).

This will give you the feasible region, which is bounded by the polygon formed by these vertices.

Do you need any further details or have any questions?

### Related Questions:
1. What method is used to solve systems of linear inequalities graphically?
2. How do you determine the points of intersection between two lines?
3. What is the significance of the feasible region in a system of inequalities?
4. How would the graph change if the inequalities were strict (e.g., \( < \) instead of \( \leq \))?
5. How do you verify if a point lies within the feasible region?

### Tip:
Always check that the intersection points satisfy all the inequalities in the system.

Graph the feasible region for the following system of inequalities by drawing a polygon around the feasible region: x + y ≤ 9, -x + y ≤ 1, y ≥ 2.

Learn how to graph the feasible region for a system of inequalities: x + y ≤ 9, -x + y ≤ 1, and y ≥ 2. This problem covers key concepts like linear inequalities, graphing, and determining intersection points, suitable for students in Grades 9-11.

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