Linear Inequality Graph 2-Variables

Question 1
Which inequality is shown on the graph?
Responses
A	y < 4x + 5y < 4x + 5
B	y < −4xy < −4x
C	y < −4x + 5y < −4x + 5
D	y < −x + 5y < − 1 4 x + 5
 

Question 2
Which ordered pair is NOT a solution to the inequality?
Responses
A	(1, 1)(1, 1)
B	(0, 2)(0, 2)
C	(1, −2)(1, −2)
D	(−2, −2)(−2, −2)

### Question 1: Which inequality is shown on the graph?

To determine the inequality, we need to consider two aspects:

1. **Slope of the line**: The slope (\(m\)) can be identified based on how steep the line is. The given options have slopes like \( -4 \), \( -\frac{1}{4} \), and \( 4 \), so the graph must show whether it's steep or relatively flat.
   
2. **Y-intercept**: This is the point where the line crosses the y-axis. It's given in the options as \(+5\) in some of the choices.

The line must also show whether the inequality uses a strict inequality ("<" or ">") or includes the boundary ("≤" or "≥"). If the line is dashed, it represents strict inequalities ("<" or ">"). If the line is solid, it represents inclusive inequalities ("≤" or "≥").

Assuming you are asking about a specific inequality related to this graph:

- **If the line has a negative slope and crosses the y-axis at +5**, then the inequality might look like \( y < -4x + 5 \) or \( y < -\frac{1}{4}x + 5 \).
  
- **If the shading is below the line**, the inequality would use "<" (less than). If the shading is above the line, the inequality would use ">".

Given the choices, **Option C** \( y < -4x + 5 \) seems likely if the line has a negative slope and crosses the y-axis at 5, with shading below the line.

### Question 2: Which ordered pair is NOT a solution to the inequality?

For this, plug in the given ordered pairs into the inequality from **Question 1**, assuming the inequality is \( y < -4x + 5 \).

#### Checking the pairs:
1. **(1, 1)**:
   \[
   y = 1 \quad \text{and} \quad x = 1
   \]
   Plug into \( y < -4x + 5 \):
   \[
   1 < -4(1) + 5 \quad \Rightarrow \quad 1 < -4 + 5 \quad \Rightarrow \quad 1 < 1 \quad (\text{False} \ because \ 1 \not< 1)
   \]
   So, **(1, 1)** is NOT a solution.

2. **(0, 2)**:
   \[
   y = 2 \quad \text{and} \quad x = 0
   \]
   Plug into \( y < -4x + 5 \):
   \[
   2 < -4(0) + 5 \quad \Rightarrow \quad 2 < 5 \quad (\text{True})
   \]
   So, **(0, 2)** is a solution.

3. **(1, -2)**:
   \[
   y = -2 \quad \text{and} \quad x = 1
   \]
   Plug into \( y < -4x + 5 \):
   \[
   -2 < -4(1) + 5 \quad \Rightarrow \quad -2 < -4 + 5 \quad \Rightarrow \quad -2 < 1 \quad (\text{True})
   \]
   So, **(1, -2)** is a solution.

4. **(-2, -2)**:
   \[
   y = -2 \quad \text{and} \quad x = -2
   \]
   Plug into \( y < -4x + 5 \):
   \[
   -2 < -4(-2) + 5 \quad \Rightarrow \quad -2 < 8 + 5 \quad \Rightarrow \quad -2 < 13 \quad (\text{True})
   \]
   So, **(-2, -2)** is a solution.

Thus, the ordered pair **(1, 1)** is **NOT a solution**, and the correct answer for **Question 2** is **A**.

---

### Relative Questions:
1. How do you determine the slope of a line from a graph?
2. How does shading on a graph help identify the inequality?
3. What is the significance of the y-intercept in linear equations?
4. Why are dashed lines used for strict inequalities?
5. How can we verify if an ordered pair is a solution to a given inequality?

### Tip:
Always check the direction of shading on the graph to identify whether the inequality uses "<" (shading below the line) or ">" (shading above the line).

Linear Inequality Graph 2-Variables

Question 1
Which inequality is shown on the graph?
Responses
A	y < 4x + 5
y < 4x + 5
B	y < −4x
y < −4x
C	y < −4x + 5
y < −4x + 5
D	y < −x + 5
y < − 1/4 x + 5

Question 2
Which ordered pair is NOT a solution to the inequality?
Responses
A	(1, 1)(1, 1)
B	(0, 2)(0, 2)
C	(1, −2)(1, −2)
D	(−2, −2)(−2, −2)

Understand how to graph and solve linear inequalities in two variables. This problem covers important concepts like the slope-intercept form, shading regions for inequalities, and verifying solutions using ordered pairs. Ideal for students in Grades 8-10.

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