https://file.aiquickdraw.com/mathbot/file/1730019560496quytehtc.jpg

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.

Determine the feasible regions and constraints for the following inequalities:
1. x + 2y ≤ 8
2. x + y ≤ 6
3. x ≥ 0
4. y ≥ 0

Additional questions:
- 5. x + 2y ≤ 8, x + y ≤ 6, x ≥ 0, y ≥ 0
- 6. 3x + 2y ≤ 12, x + y ≤ 5, x ≥ 0, y ≥ 0

Learn to solve and graph the system of linear inequalities, including constraints x + 2y ≤ 8, x + y ≤ 6, x ≥ 0, and y ≥ 0. Explore how to determine feasible regions in the first quadrant with step-by-step explanations. Suitable for Grades 10-11.

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