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.