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.