The question in the image asks to determine the solution set for each linear inequality with two variables. Let's analyze and solve each inequality one by one:

Given Inequalities:

1. (a) $2x + y > 6$
   Rearrange for $y$: $y > -2x + 6$ This inequality describes the region above the line $y = -2x + 6$ (not including the line).

2. (b) $x + 2y < 6$
   Rearrange for $y$: $y < -\frac{1}{2}x + 3$ This inequality describes the region below the line $y = -\frac{1}{2}x + 3$.

3. (c) $3x - 4y \geq 12$
   Rearrange for $y$: $y \leq \frac{3}{4}x - 3$ This inequality describes the region below or on the line $y = \frac{3}{4}x - 3$.

4. (d) $4x - 3y \leq 12$
   Rearrange for $y$: $y \geq \frac{4}{3}x - 4$ This inequality describes the region above or on the line $y = \frac{4}{3}x - 4$.

5. (e) $-6x + 3y \geq 18$
   Rearrange for $y$: $y \geq 2x + 6$ This inequality describes the region above or on the line $y = 2x + 6$.

6. (f) $-6x + 3y \leq 18$
   Rearrange for $y$: $y \leq 2x + 6$ This inequality describes the region below or on the line $y = 2x + 6$.

7. (g) $-5x - 2y > 10$
   Rearrange for $y$: $y < -\frac{5}{2}x - 5$ This inequality describes the region below the line $y = -\frac{5}{2}x - 5$.

8. (h) $-2x - 5y < 10$
   Rearrange for $y$: $y > -\frac{2}{5}x - 2$ This inequality describes the region above the line $y = -\frac{2}{5}x - 2$.

Steps to Determine the Solution Set:

1. For each inequality, graph the corresponding line.
2. Shade the region specified by the inequality (above, below, or including the line).
3. The solution set for all inequalities combined is the intersection of the shaded regions if multiple inequalities are considered.

Would you like me to graph these inequalities for visual clarity, or do you need a detailed solution for a specific inequality? Let me know!

Follow-up Questions:

1. How do you identify the boundary line of a linear inequality?
2. What is the role of testing a point (e.g., $(0, 0)$) in determining the solution region?
3. Can a solution set of inequalities be empty? Why?
4. How does flipping the inequality sign affect the solution?
5. What are real-world applications of systems of linear inequalities?

Tip:

When solving inequalities, always double-check if the inequality is strict ($>, <$) or includes equality ($\geq, \leq$) to correctly interpret the solution region!