Asked by k.
How is the graph of a system of linear inequalities related to the solutions of the system of the inequalities?
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Answered by
GPT-5 mini
AI
Each linear inequality defines a half‑plane (one side of its boundary line). The graph of a system of linear inequalities is the intersection of all those half‑planes — that intersection is exactly the set of all solutions of the system.
Key points:
- To graph a system, graph the boundary line for each inequality (use a solid line for ≤ or ≥, a dashed line for < or >) and shade the half‑plane that satisfies the inequality.
- The solution set is the region where all shadings overlap (the common region). That region may be bounded (a polygon), unbounded, a line or point, or empty (no solution).
- Any point inside the overlapping region satisfies every inequality; any point outside fails at least one.
Example: For x + y ≤ 4 and x − y ≥ 1, graph the lines x + y = 4 and x − y = 1 (solid), shade the side x + y ≤ 4 and the side x − y ≥ 1. The solution set is the overlapping shaded region.
Key points:
- To graph a system, graph the boundary line for each inequality (use a solid line for ≤ or ≥, a dashed line for < or >) and shade the half‑plane that satisfies the inequality.
- The solution set is the region where all shadings overlap (the common region). That region may be bounded (a polygon), unbounded, a line or point, or empty (no solution).
- Any point inside the overlapping region satisfies every inequality; any point outside fails at least one.
Example: For x + y ≤ 4 and x − y ≥ 1, graph the lines x + y = 4 and x − y = 1 (solid), shade the side x + y ≤ 4 and the side x − y ≥ 1. The solution set is the overlapping shaded region.
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