Ashley graphed a system of linear inequalities on the coordinate plane.
 


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

A.

y > 2x + 4y > 2x + 4

y ≤ ‒x + 2y ≤ ‒x + 2

B.

2x ‒ y ≥ 42x ‒ y ≥ 4

y < ‒x + 2y < ‒x + 2

C.

y ≥ 2x + 4y ≥ 2x + 4

‒x + y < 2‒x + y < 2

D.

‒2x + y ≥ 4‒2x + y ≥ 4

x + y < 2x + y < 2


It appears that we're looking for the correct system of linear inequalities that represents the shaded region on Ashley's graph. To determine the best choice, let's break down each option and analyze what each inequality represents.

### Step-by-Step Analysis of Each Option

1. **Inequality Type and Region:**
   - **Greater than/Greater than or Equal (e.g., \( y > 2x + 4 \) or \( y \geq 2x + 4 \)):** Shading would be above the line.
   - **Less than/Less than or Equal (e.g., \( y < -x + 2 \) or \( y \leq -x + 2 \)):** Shading would be below the line.

2. **Slopes and Y-intercepts:**
   - Look for the slope and y-intercept of each inequality to determine which region they affect. Then, match this to Ashley’s graph.

Let's go through each option to see what regions they would represent:

### Option A
1. \( y > 2x + 4 \): Shading above the line with slope 2 and y-intercept 4.
2. \( y \leq -x + 2 \): Shading below the line with slope -1 and y-intercept 2.

### Option B
1. \( 2x - y \geq 4 \): This rearranges to \( y \leq 2x - 4 \), so the shading is below the line with slope 2 and y-intercept -4.
2. \( y < -x + 2 \): Shading below the line with slope -1 and y-intercept 2.

### Option C
1. \( y \geq 2x + 4 \): Shading above the line with slope 2 and y-intercept 4.
2. \( -x + y < 2 \): Rearranges to \( y < x + 2 \), shading below the line with slope 1 and y-intercept 2.

### Option D
1. \( -2x + y \geq 4 \): Rearranges to \( y \geq 2x + 4 \), so the shading is above the line with slope 2 and y-intercept 4.
2. \( x + y < 2 \): Rearranges to \( y < -x + 2 \), shading below the line with slope -1 and y-intercept 2.

### Conclusion
Both **Options C and D** describe systems that could represent the shaded area, depending on which lines match Ashley's graph. The difference is subtle between them, but if you confirm which lines are solid (for \(\geq\) or \(\leq\)) and which are dashed (for \(>\) or \(<\)), we can identify the best match.

Would you be able to describe the lines or upload a photo of the graph? This will ensure accuracy. 

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Would you like more details, or do you have any questions? 

1. How can we interpret the slope and y-intercept in inequalities?
2. How does shading direction work in linear inequalities?
3. What does it mean for a line to be solid or dashed in graphing inequalities?
4. How do we rearrange inequalities to better understand them?
5. Why is the form \( y = mx + b \) useful in graphing?

**Tip:** When given a system of inequalities, graph each line separately, shade the corresponding region, and then find the overlapping area.

Ashley graphed a system of linear inequalities on the coordinate plane. Which system of linear inequalities is best represented by the graph?

A. y > 2x + 4, y ≤ ‒x + 2

B. 2x ‒ y ≥ 4, y < ‒x + 2

C. y ≥ 2x + 4, ‒x + y < 2

D. ‒2x + y ≥ 4, x + y < 2

Explore how to identify a system of linear inequalities based on a graph with this step-by-step solution. This problem covers key concepts in graphing inequalities and analyzing regions based on inequality signs and boundary lines, suitable for students in Grades 8-10.

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