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.
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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.
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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.
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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.
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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).