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?

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.