6-6 Systems of inequalities Updated

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What is a system of inequalities?

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A system of inequalities is a set of two or more inequalities that share the same variables. The solution is the set of all points that satisfy all inequalities in the system.

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How do you graph a linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a dashed or solid line (depending on whether the inequality is strict or inclusive). Then, shade the region that satisfies the inequality.

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What does it mean if a point is a solution to a system of inequalities?

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A point is a solution to a system of inequalities if it satisfies all inequalities in the system, meaning it lies in the shaded region of the graph.

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What is the difference between a strict and non-strict inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., ≤ or ≥) includes the boundary line.

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How can you determine if a point is a solution to a system of inequalities?

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To determine if a point is a solution, substitute the point's coordinates into each inequality. If the point satisfies all inequalities, it is a solution.

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What does it mean if a system of inequalities has no solution?

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A system of inequalities has no solution if there is no point that satisfies all inequalities, often resulting in parallel lines that do not intersect.

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What is the significance of the shaded region in the graph of a system of inequalities?

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The shaded region represents all possible solutions to the system of inequalities, where any point within this region satisfies all inequalities.

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How do you find the intersection of two inequalities?

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To find the intersection of two inequalities, graph both inequalities and identify the region where the shaded areas overlap. This overlapping region represents the solution set.

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What is a feasible region in the context of linear programming?

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A feasible region is the set of all possible points that satisfy a system of inequalities, representing all potential solutions to the problem.

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What is the role of boundary lines in graphing inequalities?

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Boundary lines separate the shaded region (solutions) from the non-shaded region (non-solutions). They are drawn based on the corresponding linear equations.