The image you've uploaded contains a set of inequalities that are likely intended for graphical representation as a system of linear inequalities. Here is a breakdown of the inequalities listed:

1. First Problem Set (Questions 1-4):

   • $x + 2y \leq 8$
   • $x + y \leq 6$
   • $x \geq 0$
   • $y \geq 0$

   These inequalities define a region in the first quadrant of the coordinate plane, bounded by the lines $x + 2y = 8$ and $x + y = 6$. The conditions $x \geq 0$ and $y \geq 0$ ensure that the feasible region remains in the positive quadrant.

2. Second Problem Set (Question 5):

   • This appears to be a combination of the inequalities in Questions 1-4:
   • $x + 2y \leq 8$
   • $x + y \leq 6$
   • $x \geq 0$
   • $y \geq 0$

   This is likely intended for defining the feasible region and then analyzing it further, possibly to find vertices or optimize a function.

3. Third Problem Set (Question 6):

   • $3x + 2y \leq 12$
   • $x + y \leq 5$
   • $x \geq 0$
   • $y \geq 0$

   This set defines another region in the first quadrant, with boundaries determined by the lines $3x + 2y = 12$ and $x + y = 5$, again constrained by $x \geq 0$ and $y \geq 0$.

Would you like a detailed solution for finding the feasible regions, or any specific analysis of these inequalities?

Related Questions:

1. How do you graphically represent a system of linear inequalities?
2. How can we determine the vertices of the feasible region?
3. What methods can be used to optimize a function within a feasible region?
4. How does changing inequality signs affect the feasible region?
5. How would these inequalities change if an objective function is introduced?

Tip: When graphing inequalities, it’s helpful to first draw the boundary lines (using equality) and then shade the region that satisfies the inequality direction.