Question
Which system of inequalities is represented by the graph?
Responses
{y≥1y−x>0
System of inequalities. First inequality. y greater than or equal to 1. Second inequality. y minus x greater than 0
{y≤1y−x<0
System of inequalities. First inequality. y less than or equal to 1. Second inequality. y minus x less than 0
{y≤1y−x>0
System of inequalities. First inequality. y less than or equal to 1. Second inequality. y minus x greater than 0
{y≥1y−x<0
System of inequalities. First inequality. y greater than or equal to 1. Second inequality. y minus x less than 0
Coordinate graph showing a range of negative five to five on the x and y axes. A dotted line crosses the points begin ordered pair zero comma zero end ordered pair and begin ordered pair one comma one end ordered pair. A solid line crosses the points begin ordered pair one comma one end ordered pair and begin ordered pair three comma one end ordered pair. The intersection of the areas below both lines is shaded.
Responses
{y≥1y−x>0
System of inequalities. First inequality. y greater than or equal to 1. Second inequality. y minus x greater than 0
{y≤1y−x<0
System of inequalities. First inequality. y less than or equal to 1. Second inequality. y minus x less than 0
{y≤1y−x>0
System of inequalities. First inequality. y less than or equal to 1. Second inequality. y minus x greater than 0
{y≥1y−x<0
System of inequalities. First inequality. y greater than or equal to 1. Second inequality. y minus x less than 0
Coordinate graph showing a range of negative five to five on the x and y axes. A dotted line crosses the points begin ordered pair zero comma zero end ordered pair and begin ordered pair one comma one end ordered pair. A solid line crosses the points begin ordered pair one comma one end ordered pair and begin ordered pair three comma one end ordered pair. The intersection of the areas below both lines is shaded.
Answers
GPT-4o mini
Based on the description of the graph and the provided options, let's analyze the key points:
1. **Solid line at y = 1**: This implies that the first inequality must be \(y \geq 1\) since it is a line where points on or above this line are included in the solution.
2. **Dotted line with a positive slope passing through (0,0) and (1,1)**: The equation of this line can be written as \(y = x\), which means the area below this line (where \(y < x\) is true) must be shaded as the dotted line indicates that it is not included in the solution. Therefore, this corresponds to the inequality \(y - x < 0\).
Combining both observations, the system of inequalities that the graph represents is:
**{y ≥ 1, y − x < 0}**
So the correct response from the options you provided is:
**{y ≥ 1, y − x < 0}**
1. **Solid line at y = 1**: This implies that the first inequality must be \(y \geq 1\) since it is a line where points on or above this line are included in the solution.
2. **Dotted line with a positive slope passing through (0,0) and (1,1)**: The equation of this line can be written as \(y = x\), which means the area below this line (where \(y < x\) is true) must be shaded as the dotted line indicates that it is not included in the solution. Therefore, this corresponds to the inequality \(y - x < 0\).
Combining both observations, the system of inequalities that the graph represents is:
**{y ≥ 1, y − x < 0}**
So the correct response from the options you provided is:
**{y ≥ 1, y − x < 0}**