Graph the solution set.

*x* − *y*

 ≥ 

1

*x* + *y*

 ≤ 

3

To graph the solution set of the system of inequalities:

1. **x - y ≥ 1**
2. **x + y ≤ 3**

### Step-by-step solution:

1. **Graph the boundary lines**:
   - For **x - y = 1**, rearrange this to **y = x - 1**. This is a line with a slope of 1 and y-intercept at -1.
   - For **x + y = 3**, rearrange this to **y = -x + 3**. This is a line with a slope of -1 and y-intercept at 3.

2. **Shading the regions**:
   - For **x - y ≥ 1**: The inequality indicates that we should shade above the line **y = x - 1** (since it's a "greater than" inequality).
   - For **x + y ≤ 3**: The inequality indicates that we should shade below the line **y = -x + 3** (since it's a "less than or equal to" inequality).

3. **Find the intersection points**: 
   - To find where the lines intersect, solve the system of equations:
     - \( y = x - 1 \)
     - \( y = -x + 3 \)

     Set them equal to each other:  
     \( x - 1 = -x + 3 \)  
     \( 2x = 4 \)  
     \( x = 2 \).  
     
     Plug \( x = 2 \) into \( y = x - 1 \):  
     \( y = 2 - 1 = 1 \).

     So, the lines intersect at the point \( (2, 1) \).

4. **Graph the solution set**: 
   - The region where the two shaded areas overlap is the solution set. This is the region bounded by the lines and shaded accordingly.

I can generate the graph to visualize this. Would you like me to do so?

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**Related questions**:
1. How do we solve systems of inequalities with more than two inequalities?
2. What are other methods to find the intersection of two lines algebraically?
3. How does changing the direction of the inequality (≥ to ≤) affect the solution region?
4. What happens when an inequality involves strict inequality (>, <)?
5. How can we determine if a point is in the solution set?

**Tip**: Always test a point (such as the origin) to confirm the direction of shading in inequality graphs.

Graph the solution set for the system of inequalities: x − y ≥ 1 and x + y ≤ 3.

This page explains how to graph the solution set for the system of inequalities x − y ≥ 1 and x + y ≤ 3. It includes step-by-step instructions for graphing boundary lines, finding intersection points, and determining the solution region. Ideal for students in Grades 9-11 learning about linear inequalities.

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