Question
Which system of linear inequalities is represented by the graph?
y > x – 2 and y < x + 1
y < x – 2 and y > x + 1
y < x – 2 and y > x + 1
y > x – 2 and y < x + 1
Answers
GPT-4o mini
To determine which system of linear inequalities is represented by the graph, we need to analyze the lines and the shaded regions of the graph.
1. The line \(y = x - 2\) is a line with a slope of 1 that intersects the y-axis at -2. If the area above this line is shaded, it represents \(y > x - 2\). If the area below it is shaded, it represents \(y < x - 2\).
2. The line \(y = x + 1\) is also a line with a slope of 1 that intersects the y-axis at +1. If the area above this line is shaded, it represents \(y > x + 1\), and if the area below it is shaded, it indicates \(y < x + 1\).
Now, examining the statements given:
- **y > x - 2 and y < x + 1**: This would represent a region bounded above by the line \(y = x + 1\) and below by the line \(y = x - 2\).
- **y < x - 2 and y > x + 1**: This is unlikely as one cannot be simultaneously above \((y > x + 1)\) and below \((y < x - 2)\) between these two lines.
- **y < x - 2 and y > x + 1**: Similar to the previous statement, this is impossible as one cannot be both below \(y = x - 2\) and above \(y = x + 1\).
- **y > x - 2 and y < x + 1**: This is feasible if the graph has a region where points are above the line \(y = x - 2\) and below the line \(y = x + 1\).
To conclude, if the graph shows shading between the lines \(y = x - 2\) and \(y = x + 1\) (the region that is above the first line and below the second), then the correct choice would be:
**y > x - 2 and y < x + 1**.
1. The line \(y = x - 2\) is a line with a slope of 1 that intersects the y-axis at -2. If the area above this line is shaded, it represents \(y > x - 2\). If the area below it is shaded, it represents \(y < x - 2\).
2. The line \(y = x + 1\) is also a line with a slope of 1 that intersects the y-axis at +1. If the area above this line is shaded, it represents \(y > x + 1\), and if the area below it is shaded, it indicates \(y < x + 1\).
Now, examining the statements given:
- **y > x - 2 and y < x + 1**: This would represent a region bounded above by the line \(y = x + 1\) and below by the line \(y = x - 2\).
- **y < x - 2 and y > x + 1**: This is unlikely as one cannot be simultaneously above \((y > x + 1)\) and below \((y < x - 2)\) between these two lines.
- **y < x - 2 and y > x + 1**: Similar to the previous statement, this is impossible as one cannot be both below \(y = x - 2\) and above \(y = x + 1\).
- **y > x - 2 and y < x + 1**: This is feasible if the graph has a region where points are above the line \(y = x - 2\) and below the line \(y = x + 1\).
To conclude, if the graph shows shading between the lines \(y = x - 2\) and \(y = x + 1\) (the region that is above the first line and below the second), then the correct choice would be:
**y > x - 2 and y < x + 1**.