Graphing systems of Inequalities

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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 for a point to be 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 significance of the shaded region in graphing inequalities?

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The shaded region represents all possible solutions to the inequality. Points within this region satisfy the inequality, while points outside do not.

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

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A strict inequality (< or >) does not include the boundary line, while a non-strict inequality (≤ 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 happens when two inequalities are parallel?

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If two inequalities are parallel and do not overlap, there is no solution. However, if they are parallel and the shaded regions overlap, there can be infinitely many solutions.

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What does it mean if a point lies on the boundary line of an inequality?

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If a point lies on the boundary line of an inequality, it is a solution if the inequality is non-strict (≤ or ≥). If the inequality is strict (< or >), it is not a solution.

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What is the solution set of a system of inequalities?

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The solution set is the collection of all points that satisfy all inequalities in the system, represented by the overlapping shaded regions on the graph.

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How do you identify a point that is NOT part of the solution set?

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A point that does not satisfy at least one of the inequalities in the system is not part of the solution set.