Systems of Linear Inequalities

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What is a system of linear inequalities?

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A system of linear inequalities is a set of two or more inequalities with 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 system of linear inequalities?

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To graph a system of linear inequalities, first graph each inequality as if it were an equation. Use a dashed line for inequalities that do not include equality (e.g., < or >) and a solid line for those that do (e.g., ≤ or ≥). Then, shade the appropriate region for each inequality, and the solution is where the shaded regions overlap.

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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 overlapping shaded region of the graph.

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What is the significance of the boundary line in a linear inequality?

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The boundary line represents the values where the inequality changes from true to false. Points on the line may or may not be included in the solution set, depending on whether the inequality is strict or inclusive.

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What is the difference between a strict inequality and an inclusive inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while an inclusive inequality (e.g., ≤ or ≥) includes the boundary line in the solution set.

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How can you determine if a point is a solution to a linear inequality?

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To determine if a point is a solution, substitute the coordinates of the point into the inequality. If the resulting statement is true, then the point is a solution.

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What is the graphical representation of the solution to a system of linear inequalities?

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The graphical representation is the region where the shaded areas of all inequalities overlap, indicating all possible solutions.

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What does it mean for two inequalities to be independent?

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Two inequalities are independent if the solution set of one does not affect the solution set of the other, meaning they can be satisfied simultaneously without restriction.

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How do you find the intersection of two linear inequalities?

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To find the intersection, graph both inequalities and identify the region where their shaded areas overlap. This overlapping region represents the intersection.

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

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A feasible region is the area on a graph that satisfies all constraints represented by the inequalities in a system.