To analyze the system of inequalities and determine the correct graph, let's carefully break down the equations provided:

System of Inequalities:

1. $x + y \leq 1$
2. $x - y \leq 2$

Steps to Solve:

1. Graph each inequality:

   • For $x + y = 1$: Rearrange as $y = -x + 1$. This is a straight line with a slope of $-1$ and a y-intercept of $1$.
   • For $x - y = 2$: Rearrange as $y = x - 2$. This is a straight line with a slope of $1$ and a y-intercept of $-2$.

2. Shading the regions:

   • For $x + y \leq 1$: Shade below the line $y = -x + 1$.
   • For $x - y \leq 2$: Shade below the line $y = x - 2$ (or equivalently, to the right of the line $x - y = 2$).

3. Identify the solution region:

   • The solution to the system is the overlapping shaded region of both inequalities.

Determining the Correct Graph:

From the provided graphs, the solution corresponds to the area where the two shaded regions overlap. This area is the triangular region bounded by the two lines.

Looking at the uploaded images, the correct graph should show shading in the region below both lines, and not extend to areas that violate either inequality. Let me confirm this by checking the visual match from the uploads.

If you'd like, I can extract more detailed observations from the images. Would you like further clarification? Let me know!

Here are 5 related questions to deepen your understanding:

1. How do you find the intersection point of the two lines in a system of inequalities?
2. Why is the boundary line solid for these inequalities?
3. How can you check whether a point lies in the solution region?
4. What is the difference between $<$ and $\leq$ in shading graphs of inequalities?
5. How can inequalities be used in real-world applications like optimization?

Tip: Always verify the shading by testing a point (e.g., $(0,0)$) in the inequalities to ensure the correct region is shaded.