The image shows a graph with two inequalities, and the task is to match the correct system of inequalities to the shaded region on the graph.

Analysis:

1. Understanding the Graph:

   • The graph shows two regions: one shaded in pink, and the other in purple.
   • The boundary line, $y = x$, separates the two regions.
   • The line is solid, indicating that the inequality is either $\leq$ or $\geq$.
   • The pink region lies above the line $y = x$, indicating $y \geq x$.
   • The purple region lies to the left of the vertical line $x = -1$.

2. Checking the Given Options:

   • We need to determine which option correctly represents the system of inequalities that describes the shaded region.

Steps:

• The pink region is $y \geq x$.
• The vertical boundary at $x = -1$ corresponds to $x \leq -1$ for the purple region.

Answer:

The correct system of inequalities that matches the graph is: $\textbf{y} \geq \textbf{x} \quad \text{and} \quad \textbf{x} \leq -1$

This corresponds to the top right option in the image.

Would you like a more detailed explanation or have any questions?

Related Questions:

1. What does the solid line in the graph of an inequality represent?
2. How do you determine which region to shade when graphing inequalities?
3. What changes in the graph if the inequality is strict (e.g., $<$ or $>$)?
4. How do you interpret a system of inequalities geometrically?
5. What is the significance of the boundary lines in the context of inequalities?

Tip:

When analyzing systems of inequalities, always check if the boundary lines are solid or dashed to correctly determine the inclusive or exclusive nature of the inequalities.