Solve each system of inequalities by graphing. y - x < 3, y - x ≥ 2

To solve the system of inequalities by graphing, we'll graph each inequality and find the region where the solutions overlap. Here are the steps:

### Inequality 1: \( y - x < 3 \)
1. **Rewrite the inequality** in slope-intercept form (i.e., \( y = mx + b \)):
   \[
   y < x + 3
   \]
2. **Graph the boundary line** \( y = x + 3 \). This line will have a slope of 1 and a y-intercept of 3.
3. **Determine the region** to shade: Since the inequality is \( y < x + 3 \), shade the region **below** the line.

### Inequality 2: \( y - x \geq 2 \)
1. **Rewrite the inequality** in slope-intercept form:
   \[
   y \geq x + 2
   \]
2. **Graph the boundary line** \( y = x + 2 \). This line will also have a slope of 1 and a y-intercept of 2.
3. **Determine the region** to shade: Since the inequality is \( y \geq x + 2 \), shade the region **above** the line.

### Graphing and Finding the Solution
1. **Graph both boundary lines**:
   - \( y = x + 3 \) (dashed line, because the inequality is strict)
   - \( y = x + 2 \) (solid line, because the inequality is inclusive)
2. **Shade the appropriate regions**:
   - Below the line \( y = x + 3 \)
   - Above the line \( y = x + 2 \)
3. **Identify the overlapping region**: The solution set is the region where the shaded areas overlap.

### Solution
The solution is the region between the lines \( y = x + 2 \) and \( y = x + 3 \), where:
- \( y > x + 2 \) (below the dashed line)
- \( y \leq x + 3 \) (above the solid line)

This region forms a strip parallel to the lines, above \( y = x + 2 \) and below \( y = x + 3 \).

Would you like me to graph this for you or provide further details?

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Here are 5 related questions:

1. How do you solve systems of linear inequalities with more than two inequalities?
2. What is the difference between a strict inequality and an inclusive inequality?
3. How does the slope of a line affect the solution of a system of inequalities?
4. What happens when the inequalities in a system have no overlapping region?
5. Can you explain how to find the vertices of the solution region for a system of linear inequalities?

**Tip:** When graphing inequalities, always pay attention to whether the boundary line should be dashed or solid—this indicates whether the line itself is part of the solution set.

Learn how to solve systems of linear inequalities by graphing. This problem involves graphing two inequalities, y - x < 3 and y - x ≥ 2, finding the overlapping region, and identifying the solution set. Suitable for students in Grades 9-12.

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