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To solve this problem, we need to graph the feasible region for the system of inequalities provided:

1. \( x + y \leq 8 \)
2. \(-x + y \geq 4 \)
3. \(x \geq 0 \)
4. \(y \geq 0 \)

Let's analyze and plot these inequalities one by one:

### Step 1: Graph the inequality \(x + y \leq 8\)
- Rearrange to the slope-intercept form: \(y \leq -x + 8\).
- Plot the line \(y = -x + 8\), which has a y-intercept at \( (0, 8) \) and an x-intercept at \( (8, 0) \).
- Shade the area below this line.

### Step 2: Graph the inequality \(-x + y \geq 4\)
- Rearrange to the slope-intercept form: \(y \geq x + 4\).
- Plot the line \(y = x + 4\), which has a y-intercept at \( (0, 4) \) and an x-intercept at \( (-4, 0) \).
- Shade the area above this line.

### Step 3: Graph the inequalities \(x \geq 0\) and \(y \geq 0\)
- These inequalities indicate that the feasible region lies in the first quadrant (where both x and y are positive).

### Step 4: Identify the feasible region
- The feasible region will be the overlapping area that satisfies all these inequalities.

### Step 5: Draw the polygon
- The vertices of this polygon are determined by the intersections of these lines:
  - The intersection of \(x + y = 8\) and \(-x + y = 4\).
  - The intersections with the x-axis and y-axis at relevant points.

#### Intersection points to plot:
- The lines \(x + y = 8\) and \(-x + y = 4\) intersect at \( (2, 6) \).
- The line \(x + y = 8\) intersects the x-axis at \( (8, 0) \) and the y-axis at \( (0, 8) \).
- The line \(-x + y = 4\) intersects the y-axis at \( (0, 4) \).

Using these points:
- The vertices of the feasible region polygon are \( (0, 4) \), \( (2, 6) \), \( (8, 0) \), and possibly another relevant point.

Would you like a more detailed step-by-step guide on how to draw the polygon or any clarifications?

### Related Questions
1. What is the method for finding the intersection of two lines algebraically?
2. How do you identify the vertices of a feasible region in a graph of inequalities?
3. What does it mean for a region to be "feasible" in linear programming?
4. How does the slope of a line relate to its equation in slope-intercept form?
5. What are the boundary conditions when solving inequalities graphically?

### Tip
Always check which side of the boundary line satisfies the inequality by choosing a test point that is not on the line.

Graph the feasible region for the following system of inequalities:
1. x + y ≤ 8
2. -x + y ≥ 4
3. x ≥ 0
4. y ≥ 0

Learn how to graph the feasible region for a system of linear inequalities, including x + y ≤ 8 and -x + y ≥ 4. The solution explores intersection points, graphing, and shading to represent the region of overlap. Suitable for high school students in Grades 9-12.

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