https://file.aiquickdraw.com/mathbot/file/1725896383409hx4gdh3d.jpg

To match the system of inequalities to the graph, let's analyze the lines and shaded regions:

### Step-by-Step Analysis

1. **Identify the Equations of the Lines:**

   - The graph shows two lines: one blue and one red.
   - The blue line passes through the points \((-4, 0)\) and \((0, 4)\). The slope (\(m\)) is:
     \[
     m = \frac{4 - 0}{0 - (-4)} = \frac{4}{4} = 1.
     \]
     The y-intercept (\(b\)) is 4. Therefore, the equation of the blue line is:
     \[
     y = x + 4.
     \]

   - The red line passes through the points \((0, 0)\) and \((2, 4)\). The slope (\(m\)) is:
     \[
     m = \frac{4 - 0}{2 - 0} = 2.
     \]
     The y-intercept (\(b\)) is 0. Therefore, the equation of the red line is:
     \[
     y = 2x.
     \]

2. **Determine the Inequalities:**

   - The blue line separates the purple and blue shaded regions. The shading is below the line, so the inequality is:
     \[
     y \leq x + 4.
     \]

   - The red line separates the red and purple shaded regions. The shading is to the left of the line, so the inequality is:
     \[
     y \geq 2x.
     \]

3. **Combine the Inequalities:**

   - The overlapping purple region represents the solution to the system:
     \[
     \begin{cases}
     y \leq x + 4, \\
     y \geq 2x.
     \end{cases}
     \]

### Conclusion

The system of inequalities that matches the graph is:
\[
\begin{cases}
y \leq x + 4, \\
y \geq 2x.
\end{cases}
\]

Would you like more details or have any questions?

### Further Questions:
1. How would the graph change if the inequality \(y \leq x + 4\) became \(y < x + 4\)?
2. What would the shaded region look like if the inequality \(y \geq 2x\) became \(y > 2x\)?
3. Can you find the coordinates of the intersection point of the two lines?
4. How would the solution set change if we reversed both inequalities?
5. What are the implications of graphing inequalities with different slopes?

**Tip:** When graphing inequalities, always check which side of the line is shaded by substituting a test point (like the origin, if not on the line).

Which of the following systems of inequalities matches the graph shown?

Analyze and solve the system of inequalities matching the given graph. Understand key algebraic concepts such as slope, linear equations, and graphing inequalities. Suitable for students in Grades 8-10.

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