To determine which graph represents the solution to the system of inequalities:

1. Identify each inequality and rewrite them in slope-intercept form (if necessary) to understand how they should appear on a graph.

   • $x + y < 1$

     • Rewrite as $y < -x + 1$.
     • This inequality has a boundary line with a slope of -1 and a y-intercept at $(0, 1)$.
     • Since it is a strict inequality ( $<$ ), the boundary line will be dashed, and we shade below this line.

   • $x - y \leq 2$

     • Rewrite as $y \geq x - 2$.
     • This inequality has a boundary line with a slope of 1 and a y-intercept at $(0, -2)$.
     • Since it is a non-strict inequality ( $\leq$ ), the boundary line will be solid, and we shade above this line.

2. Analyze each graph in the options:

   • We need to look for a graph where:
     • The line representing $y = -x + 1$ is dashed, with shading below it.
     • The line representing $y = x - 2$ is solid, with shading above it.
     • The shaded region should be the intersection of these two shaded areas.

Based on this analysis, the second graph (second from the top) correctly shows the solution set:

• The dashed line corresponds to $y < -x + 1$.
• The solid line corresponds to $y \geq x - 2$.
• The shading overlaps in the correct region between these lines.

Answer: The second graph is the correct representation.

Would you like more details on any step, or have any questions?

Here are some related questions:

1. How do we determine which side of the line to shade for inequalities?
2. What is the difference between solid and dashed boundary lines in graphing inequalities?
3. How do you find the intersection of two inequalities on a graph?
4. Why is it necessary to rewrite inequalities in slope-intercept form for graphing?
5. How would the solution region change if the inequalities were flipped?

Tip: When solving systems of inequalities, always check both the line type (solid or dashed) and shading direction for accuracy in graphing.