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The task here is to graph the system of inequalities given:

1. \( y \leq x - 3 \)
2. \( y \geq x + 1 \)

Let's go through the steps for graphing this system:

### Step 1: Graph the Boundary Lines
1. **For \( y = x - 3 \)**: 
   - This line has a slope of \( 1 \) and a y-intercept at \( -3 \).
   - Plot the y-intercept (0, -3) and use the slope to find other points, such as (1, -2) and (-1, -4).
   - Since the inequality is \( y \leq x - 3 \), draw a solid line to include points on the line.

2. **For \( y = x + 1 \)**:
   - This line has a slope of \( 1 \) and a y-intercept at \( 1 \).
   - Plot the y-intercept (0, 1) and use the slope to find other points, such as (1, 2) and (-1, 0).
   - Since the inequality is \( y \geq x + 1 \), also draw a solid line to include points on this line.

### Step 2: Shade the Appropriate Regions
1. For \( y \leq x - 3 \), shade below the line \( y = x - 3 \).
2. For \( y \geq x + 1 \), shade above the line \( y = x + 1 \).

### Step 3: Find the Solution Region
The solution to the system is the overlapping shaded area where both inequalities are satisfied. 

Would you like a sketch or any further explanation on the steps? 

### Relative Questions
1. How do you determine which side to shade for each inequality?
2. What happens to the graph if you change the inequality from \( \leq \) to \( < \)?
3. How would the solution change if the slope of one of the lines was altered?
4. How does the graph change if both inequalities are reversed?
5. What if there was no overlapping region; how would that be represented?

### Tip
Always check a test point in the final shaded region to ensure it satisfies both inequalities.

Graph the system of inequalities:
1. y ≤ x - 3
2. y ≥ x + 1

Learn how to graph the system of inequalities y ≤ x - 3 and y ≥ x + 1 with step-by-step instructions. Understand key concepts like graphing linear inequalities, shading solution regions, and identifying intersections of inequalities. Suitable for students in Grades 8-10.

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