Aubrey graphed the following system of linear inequalities.
 
y < ‒x + 1
x − y ≤ 2

Which graph best represents the solution set to this system of inequalities?
A.
B.
C.
D.
3b. 
Which point is NOT a solution to the system?
A.(‒4, ‒5)
B.(0, 0)
C.(‒3, 1)
D.(‒5, 6)

To analyze the system of linear inequalities given by:

1. \( y < -x + 1 \)
2. \( x - y \leq 2 \)

### Step 1: Graph the inequalities

1. **First Inequality: \( y < -x + 1 \)**
   - The boundary line is \( y = -x + 1 \), which has a y-intercept of 1 and a slope of -1.
   - The region below this line (since it's a "less than" inequality) is shaded.

2. **Second Inequality: \( x - y \leq 2 \)**
   - Rearranging gives \( y \geq x - 2 \).
   - The boundary line is \( y = x - 2 \), which has a y-intercept of -2 and a slope of 1.
   - The region above this line is shaded (since it's "greater than or equal to").

### Step 2: Identify the solution region
The solution set is the area where the shaded regions of both inequalities overlap.

### Step 3: Check the points
Now, we need to determine which point is **NOT** a solution to the system:

1. **Point A: \((-4, -5)\)**
   - For \( y < -x + 1 \): \(-5 < -(-4) + 1 \rightarrow -5 < 5 \) (True)
   - For \( x - y \leq 2 \): \(-4 - (-5) \leq 2 \rightarrow 1 \leq 2\) (True)

2. **Point B: \((0, 0)\)**
   - For \( y < -x + 1 \): \(0 < -0 + 1 \rightarrow 0 < 1\) (True)
   - For \( x - y \leq 2 \): \(0 - 0 \leq 2 \rightarrow 0 \leq 2\) (True)

3. **Point C: \((-3, 1)\)**
   - For \( y < -x + 1 \): \(1 < -(-3) + 1 \rightarrow 1 < 4\) (True)
   - For \( x - y \leq 2 \): \(-3 - 1 \leq 2 \rightarrow -4 \leq 2\) (True)

4. **Point D: \((-5, 6)\)**
   - For \( y < -x + 1 \): \(6 < -(-5) + 1 \rightarrow 6 < 6\) (False)
   - For \( x - y \leq 2 \): \(-5 - 6 \leq 2 \rightarrow -11 \leq 2\) (True)

### Conclusion
The point that is **NOT** a solution to the system is **D. \((-5, 6)\)**.

### Next Steps
Would you like to see detailed graphing steps or have any specific questions? Here are some follow-up topics:

1. How to graph linear inequalities step by step.
2. The significance of boundary lines in inequalities.
3. How to test points for solution sets.
4. Applications of linear inequalities in real life.
5. Graphing systems of equations versus inequalities. 

**Tip:** When graphing inequalities, always start with the boundary line and determine which side to shade based on the inequality.

Aubrey graphed the following system of linear inequalities: y < -x + 1 and x - y ≤ 2. Which graph best represents the solution set to this system of inequalities? Which point is NOT a solution to the system?

Explore how to graph the system of linear inequalities y < -x + 1 and x - y ≤ 2, and identify which point is NOT a solution. This problem illustrates key concepts in graphing inequalities and determining solution sets, suitable for Grades 8-10.

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