Let's break down the graph to find which system of inequalities produces the shaded region.

Step 1: Analyze the boundary lines.

• There are two boundary lines in the graph:

  1. A solid line with a negative slope. This line seems to be $y = -x + 4$.

     • Since the line is solid, the inequality includes the boundary, i.e., $y \leq -x + 4$.
     • The region is shaded below this line, confirming that the inequality involves $\leq$.

  2. A dashed line with a positive slope. This line appears to be $y = x + 2$.

     • The line is dashed, meaning the inequality does not include the boundary, i.e., $y < x + 2$.
     • The region is shaded below this line, so the inequality is $y < x + 2$.

Step 2: Identify the region.

The region shaded in the graph is below both lines. This implies that both inequalities must hold simultaneously.

Step 3: Conclusion.

From the graph, the system of inequalities representing the shaded region is:

• $y \leq -x + 4$
• $y < x + 2$

Now, checking the options:

• Option C: $y \leq -x + 4$ and $y < x + 2$ is the correct answer.

Would you like further clarification, or do you have any questions?

Here are 5 related questions you can explore:

1. How do you determine the equation of a line from a graph?
2. What is the difference between dashed and solid lines in graphing inequalities?
3. How do you graph a system of inequalities on the coordinate plane?
4. How would the region change if the inequality was $y \geq -x + 4$?
5. What happens when two inequalities share the same boundary line?

Tip: Always pay attention to whether a boundary line is solid or dashed, as it indicates whether the points on the line are included in the solution!