The given problem is to find the solution set of the following system of inequalities:

1. $x + 2y \leq 8$
2. $x \leq 8$, $y \geq -2$
3. $2x - 5y \geq 10$

Solution Process:

Step 1: Analyze the inequalities

1. $x + 2y \leq 8$:

   • Rearrange for $y$: $y \leq \frac{-x + 8}{2}$
   • This represents a half-plane below the line $y = \frac{-x + 8}{2}$.

2. $x \leq 8$:

   • This inequality means that the solution is in the region to the left of the vertical line $x = 8$.

3. $y \geq -2$:

   • This means the solution is in the region above the horizontal line $y = -2$.

4. $2x - 5y \geq 10$:

   • Rearrange for $y$: $y \leq \frac{2x - 10}{5}$
   • This represents a half-plane below the line $y = \frac{2x - 10}{5}$.

Step 2: Find the feasible region

The solution set is the intersection of all regions defined by these inequalities. To graph:

1. Plot the lines:
   • $x + 2y = 8$ (slope: $-1/2$).
   • $x = 8$ (vertical line).
   • $y = -2$ (horizontal line).
   • $2x - 5y = 10$ (slope: $2/5$).
2. Shade the regions based on the direction of the inequalities.
3. The feasible region is the common overlapping area that satisfies all inequalities.

Step 3: Determine vertices

Solve the intersections of the lines to find the vertices of the feasible region:

1. $x + 2y = 8$ and $2x - 5y = 10$.
2. $x + 2y = 8$ and $y = -2$.
3. $2x - 5y = 10$ and $y = -2$.
4. Check boundary points, such as $x = 8$.

Would you like me to explicitly compute the vertices or provide a visualization? Let me know if you'd like any clarification!

Related Questions:

1. What are the vertices of the feasible region?
2. How do you graph inequalities effectively on a coordinate plane?
3. How can we verify if a point satisfies all given inequalities?
4. What happens if the system of inequalities has no solution?
5. How can optimization problems use this feasible region?

Tip: Always check the direction of the inequality when graphing—mistakes can lead to incorrect feasible regions!