https://file.aiquickdraw.com/mathbot/file/1728746927351baneraes.jpg

To graph the system of inequalities and show the solution region, let’s go step by step.

The system consists of two inequalities:

1. \( y < 2x + 6 \)
2. \( y \geq -x + 4 \)

### Step 1: Graph the boundary lines
- The inequality \( y < 2x + 6 \) has a boundary line \( y = 2x + 6 \), which is a straight line with slope 2 and y-intercept 6. Since the inequality is strict ("<"), the boundary line will be **dashed**.
- The inequality \( y \geq -x + 4 \) has a boundary line \( y = -x + 4 \), which is a straight line with slope -1 and y-intercept 4. Since the inequality includes "greater than or equal to" (≥), the boundary line will be **solid**.

### Step 2: Plot the lines
- For \( y = 2x + 6 \):
  - Start by plotting the y-intercept (0, 6).
  - Use the slope (2) to find another point. From (0, 6), go up 2 units and right 1 unit, reaching (1, 8).
  - Draw a **dashed** line through these points.
  
- For \( y = -x + 4 \):
  - Start by plotting the y-intercept (0, 4).
  - Use the slope (-1) to find another point. From (0, 4), go down 1 unit and right 1 unit, reaching (1, 3).
  - Draw a **solid** line through these points.

### Step 3: Determine shading for each inequality
- For \( y < 2x + 6 \), the region will be **below** the dashed line.
- For \( y \geq -x + 4 \), the region will be **above or on** the solid line.

### Step 4: Identify the solution region
The solution region is where the two shaded regions overlap. This area will be below the dashed line \( y = 2x + 6 \) and above the solid line \( y = -x + 4 \).

I can create a visual graph for you if needed. Would you like me to generate an image with the graph based on this information?

### Relative Questions:
1. How do we determine whether to use a solid or dashed line in an inequality?
2. What is the significance of the slope in graphing these inequalities?
3. How do we know which region to shade for a given inequality?
4. What happens if the inequalities were reversed, such as \( y > 2x + 6 \)?
5. How can we confirm the points where the lines intersect algebraically?

### Tip:
Always test a point (like the origin, \( (0, 0) \), if it's not on the boundary) to help verify the correct shading for inequalities.

Graph the system: y < 2x + 6, y ≥ -x + 4

This problem explains how to graph the system of inequalities y < 2x + 6 and y ≥ -x + 4. It covers graphing boundary lines, determining shading regions, and finding the solution area for students in Grades 9-12.

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