Sketch the region that corresponds to the given inequalities. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.)
4x	 + 	3y	 ≤ 	8
4x	 − 	3y	 ≥ 	8
−y	 ≥ 	2
Inequality	Two Points 
(x, y)
 on the Graphed Line	Point 
(x, y)
 in Shaded Region
4x + 3y ≤ 8
 

 
 , 0

 

 

0, 
 

 
 

 
  

 
4x − 3y ≥ 8
 

 
 , 0

 

 

0, 
 

 
 

 
  

 
−y ≥ 2
 

0, 
 

 

 

1, 
 

 
 

 
  

 

To sketch the region corresponding to the given inequalities, let's break each inequality down:

### Inequality 1: 
\[
4x + 3y \leq 8
\]

This represents a line \( 4x + 3y = 8 \), and the inequality \( \leq \) tells us to shade the region **below** the line. To plot the line:

- Set \( x = 0 \), solve for \( y \):
\[
4(0) + 3y = 8 \implies y = \frac{8}{3} \approx 2.67
\]
So one point on the line is \( (0, 2.67) \).

- Set \( y = 0 \), solve for \( x \):
\[
4x + 3(0) = 8 \implies x = 2
\]
So another point on the line is \( (2, 0) \).

### Inequality 2: 
\[
4x - 3y \geq 8
\]

This represents a line \( 4x - 3y = 8 \), and the inequality \( \geq \) tells us to shade the region **above** the line. To plot the line:

- Set \( x = 0 \), solve for \( y \):
\[
4(0) - 3y = 8 \implies y = -\frac{8}{3} \approx -2.67
\]
So one point on the line is \( (0, -2.67) \).

- Set \( y = 0 \), solve for \( x \):
\[
4x - 3(0) = 8 \implies x = 2
\]
So another point on the line is \( (2, 0) \).

### Inequality 3: 
\[
-y \geq 2 \implies y \leq -2
\]

This inequality tells us to shade the region **below** the horizontal line \( y = -2 \).

### Graphing and Shaded Region:
- The region below \( 4x + 3y = 8 \).
- The region above \( 4x - 3y = 8 \).
- The region below \( y = -2 \).

After plotting all three lines and shading the correct regions, the solution region is where all three shaded areas overlap.

### Points:
To provide two points on each graphed line:

1. **For \( 4x + 3y = 8 \)**:
   - Two points are \( (0, 2.67) \) and \( (2, 0) \).

2. **For \( 4x - 3y = 8 \)**:
   - Two points are \( (0, -2.67) \) and \( (2, 0) \).

3. **For \( y = -2 \)**:
   - Two points are \( (0, -2) \) and any point on that line, such as \( (1, -2) \).

If you need further clarifications, feel free to ask.

---

### Additional Questions:
1. How do you identify regions for inequalities involving "greater than" or "less than"?
2. Can you explain why the intersection of the shaded regions represents the solution set?
3. How do you determine the boundaries of the solution region?
4. What happens to the solution set if one of the inequalities is strict (e.g., \( < \) or \( > \))?
5. How would you verify if a point lies in the shaded region?

### Tip:
Always test points from the shaded region to ensure they satisfy all the inequalities.

Sketch the region that corresponds to the given inequalities. (Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) 4x + 3y ≤ 8, 4x − 3y ≥ 8, −y ≥ 2. Inequality Two Points (x, y) on the Graphed Line Point (x, y) in Shaded Region.

Learn how to graph the system of inequalities 4x + 3y ≤ 8, 4x − 3y ≥ 8, and −y ≥ 2. This problem covers key concepts such as linear inequalities and graphing, and explains how to find the overlapping solution region for a system of inequalities. Suitable for students in Grades 9-12.

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