https://file.aiquickdraw.com/mathbot/file/1730394695975xb5r2he7.jpg

To determine the system of inequalities that corresponds to the shaded region in the graph, we need to analyze the boundaries and shading pattern for each line:

### Step 1: Identify Each Boundary Line
1. **Diagonal line (positive slope)**: This line passes through the origin (0,0) and appears to have a slope of 1. Its equation is:
   \[
   y = x
   \]
   Since the shaded region is above this line, the inequality is:
   \[
   y \geq x
   \]

2. **Vertical line**: This line is positioned at \( x = 4 \). Since the shaded region lies to the left of this line, the inequality is:
   \[
   x \leq 4
   \]

3. **Horizontal line**: This line is positioned at \( y = -2 \). Since the shaded region is above this line, the inequality is:
   \[
   y \geq -2
   \]

### Step 2: Write the System of Inequalities
The system of inequalities that defines the shaded triangular region is:
\[
\begin{cases}
y \geq x \\
x \leq 4 \\
y \geq -2
\end{cases}
\]

Would you like further details on interpreting graphs with inequalities, or have any specific questions?

### Additional Questions:
1. How can we determine the slope of a line by looking at its position on a graph?
2. Why does the inequality sign change direction based on the shading region?
3. What would the inequality look like if the shaded region were below a boundary line?
4. How would the system of inequalities change if the boundary lines were dashed instead of solid?
5. How can graphing inequalities help in solving real-world problems?

### Tip:
Always check the shading and boundary line direction to confirm whether each inequality is "greater than" or "less than" in graphical solutions.

The picture below shows the solution set for which system of inequalities?

Analyze a graph to determine the system of inequalities that defines a shaded region. This problem explores key concepts in algebra, including graphing inequalities and interpreting solution sets. Suitable for students in Grades 8-10.

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