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Let's solve each question step-by-step.

### Problem 11

The graph shows a shaded region in the first quadrant, indicating that the inequalities are bounded by two lines:
1. A horizontal line at \( y = 3 \), where the shading is below it, suggesting \( y \leq 3 \).
2. A line passing through the origin with a positive slope, suggesting \( y = x \) as a boundary. The shading is also below this line, which implies \( y \leq x \).

Therefore, the system of inequalities that could represent this region is:
\[
\text{C) } y \leq 3 \text{ and } y \leq x.
\]

### Problem 12

The inequalities given are:
1. \( y \geq \frac{3}{2}x \)
2. \( y \leq -2x - 5 \)

The first inequality represents a line with a positive slope, shading above it. The second inequality represents a line with a negative slope, shading below it. The solution region lies where the two shaded areas overlap, which is likely to be in the area bounded between these lines in the negative \( y \)-direction.

Looking at the answer choices, the graph that correctly represents this system is:
- **A)** because it shows shading in the region where both inequalities would overlap, which is a triangular region below these lines.

Would you like further details or have any questions about these solutions?

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Here are 5 related questions to explore further:

1. How do you determine the direction of shading for an inequality on a graph?
2. What is the significance of boundary lines being dashed or solid in inequalities?
3. How can you convert an inequality into a slope-intercept form to easily plot it?
4. What methods are available to solve a system of inequalities algebraically?
5. How do you determine which regions satisfy both inequalities when graphing?

**Tip:** When graphing inequalities, test a point (like the origin) to quickly verify the correct region for shading.

Which of the following systems of inequalities could be the one graphed in the xy-plane above? (Question 11). Which of the following graphs in the xy-plane could represent the system of inequalities above? (Question 12)

Explore how to solve systems of inequalities through graphing techniques. Includes step-by-step solutions for determining regions of overlap in inequalities, covering concepts like shading rules and line intersections. Ideal for students in Grades 10-12.

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