Question
Determine which of the graphs is the graph of a system of inequalities with no solutions.(1 point)
Option 1 shows an image with a line with an equation of y ≥ x + 2 and the other line has an equation of y ≤ x + 2
Option 2 shows an image with a line with an equation of y ≥ 2x - 1 - 2 and the other line that intersects with the first one has an equation of y ≤ 3x - 2
Option 3 shows an image with a line with the equation of y > x + 2 and the other one is y < x - 2
Option 4 shows an image with a line with the equation of y ≥ 2x + 1 and the other one has an equation of y ≥ 2x - 1
Option 1 shows an image with a line with an equation of y ≥ x + 2 and the other line has an equation of y ≤ x + 2
Option 2 shows an image with a line with an equation of y ≥ 2x - 1 - 2 and the other line that intersects with the first one has an equation of y ≤ 3x - 2
Option 3 shows an image with a line with the equation of y > x + 2 and the other one is y < x - 2
Option 4 shows an image with a line with the equation of y ≥ 2x + 1 and the other one has an equation of y ≥ 2x - 1
Answers
Answered by
GPT-4o mini
To determine which option corresponds to a system of inequalities with no solutions, we need to check if the shaded regions (solutions) from the inequalities conflict with each other.
1. **Option 1**:
- \(y \ge x + 2\): Represents the region above the line \(y = x + 2\).
- \(y \le x + 2\): Represents the region below the line \(y = x + 2\).
- **Conclusion**: There is a conflict here, as no point can simultaneously be above and below the same line, so this option has no solution.
2. **Option 2**:
- \(y \ge 2x - 1\): Represents the region above the line \(y = 2x - 1\).
- \(y \le 3x - 2\): Represents the region below the line \(y = 3x - 2\).
- **Conclusion**: If these lines intersect, it is possible for a solution to exist since the regions are separated but not conflicting.
3. **Option 3**:
- \(y > x + 2\): Represents the region above the line \(y = x + 2\).
- \(y < x - 2\): Represents the region below the line \(y = x - 2\).
- **Conclusion**: The lines are parallel and do not intersect; thus, they leave no overlap, indicating no solutions exist.
4. **Option 4**:
- \(y \ge 2x + 1\) and \(y \ge 2x - 1\): Both represent regions that are above two lines with the same slope but different y-intercepts.
- **Conclusion**: There is a possibility for solutions since both inequalities point to the area above the higher line or both satisfy their conditions.
Based on the analysis above, **Option 1** and **Option 3** indicate a system of inequalities with no solutions. However, since Option 3's regions do not intersect or overlap at all, it's the clearer example of a conflicting system.
**Final Answer: Option 3** (the graph of a system of inequalities with no solutions).
1. **Option 1**:
- \(y \ge x + 2\): Represents the region above the line \(y = x + 2\).
- \(y \le x + 2\): Represents the region below the line \(y = x + 2\).
- **Conclusion**: There is a conflict here, as no point can simultaneously be above and below the same line, so this option has no solution.
2. **Option 2**:
- \(y \ge 2x - 1\): Represents the region above the line \(y = 2x - 1\).
- \(y \le 3x - 2\): Represents the region below the line \(y = 3x - 2\).
- **Conclusion**: If these lines intersect, it is possible for a solution to exist since the regions are separated but not conflicting.
3. **Option 3**:
- \(y > x + 2\): Represents the region above the line \(y = x + 2\).
- \(y < x - 2\): Represents the region below the line \(y = x - 2\).
- **Conclusion**: The lines are parallel and do not intersect; thus, they leave no overlap, indicating no solutions exist.
4. **Option 4**:
- \(y \ge 2x + 1\) and \(y \ge 2x - 1\): Both represent regions that are above two lines with the same slope but different y-intercepts.
- **Conclusion**: There is a possibility for solutions since both inequalities point to the area above the higher line or both satisfy their conditions.
Based on the analysis above, **Option 1** and **Option 3** indicate a system of inequalities with no solutions. However, since Option 3's regions do not intersect or overlap at all, it's the clearer example of a conflicting system.
**Final Answer: Option 3** (the graph of a system of inequalities with no solutions).