Mr. Aguilar placed a graphed system of linear inequalities on the board.
 


 

Which system of linear inequalities is best represented by the graph?

A.

y ≤ 3x − 9y ≤ 3x − 9

y ≥ −45x + 7y ≥ −45x + 7

B.

y ≥ 3x − 9y ≥ 3x − 9

y ≥ −45x + 7y ≥ −45x + 7

C.

y ≥ 3x − 9y ≥ 3x − 9

y ≤ −45x + 7y ≤ −45x + 7

D.

y ≤ 3x − 9y ≤ 3x − 9

y ≤ −45x + 7y ≤ −45x + 7


To determine which system of linear inequalities matches the graph, let's analyze the inequalities provided in each option. Here’s a breakdown of what you need to look for:

### Key elements to check:
1. **Inequality Direction**:
   - \( y \leq \) or \( y \geq \) indicates if the solution region lies **below or above** the line.
   - \( y \leq \) means shading **below** the line (or on the line if equality holds).
   - \( y \geq \) means shading **above** the line (or on the line if equality holds).

2. **Slope and Y-Intercept**:
   - The inequalities are in the form \( y \leq mx + b \) or \( y \geq mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

   - For the line \( y = 3x - 9 \):
     - Slope \( m = 3 \)
     - Y-intercept \( b = -9 \)
   
   - For the line \( y = -45x + 7 \):
     - Slope \( m = -45 \)
     - Y-intercept \( b = 7 \)

### Now, let's interpret each option:

- **Option A:**
  - \( y \leq 3x - 9 \) (shading **below** this line)
  - \( y \geq -45x + 7 \) (shading **above** this line)

- **Option B:**
  - \( y \geq 3x - 9 \) (shading **above** this line)
  - \( y \geq -45x + 7 \) (shading **above** this line)

- **Option C:**
  - \( y \geq 3x - 9 \) (shading **above** this line)
  - \( y \leq -45x + 7 \) (shading **below** this line)

- **Option D:**
  - \( y \leq 3x - 9 \) (shading **below** this line)
  - \( y \leq -45x + 7 \) (shading **below** this line)

### Choosing the correct system:
The correct system will have:
- **One inequality with \( y \leq \)** and **one with \( y \geq \)** if the solution region is bounded between the two lines.

Thus, **Option A** is the most likely answer:
- \( y \leq 3x - 9 \) (shading below)
- \( y \geq -45x + 7 \) (shading above)

This option suggests the solution region is **between** the two lines, matching a typical intersection of two linear inequalities.

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Would you like a detailed graph explanation or further clarification?  
Here are five related questions to expand your understanding:

1. How do we determine if an inequality includes the boundary line?
2. What happens when the slopes of two lines are equal in a system of inequalities?
3. How do you graph a system of linear inequalities step by step?
4. How can you tell if a point lies in the solution region of a system of inequalities?
5. What does it mean if two inequalities have no overlapping solution region?

**Tip:** When graphing inequalities, use test points (like the origin) to confirm which side of the line should be shaded.

Mr. Aguilar placed a graphed system of linear inequalities on the board. Which system of linear inequalities is best represented by the graph?

A.

y ≤ 3x − 9

y ≥ −45x + 7

B.

y ≥ 3x − 9

y ≥ −45x + 7

C.

y ≥ 3x − 9

y ≤ −45x + 7

D.

y ≤ 3x − 9

y ≤ −45x + 7

Explore how to determine the correct system of linear inequalities represented by a graph. This problem covers key concepts in graphing linear inequalities, analyzing slope, y-intercept, and shading regions. Ideal for students in Grades 9-10.

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