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 linear inequalities that involve the same variables. The solution is the set of all points that satisfy all inequalities in the system.

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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 the inequalities in the system, meaning it lies in the region defined by the inequalities.

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What is the difference between a solid line and a dashed line in graphing inequalities?

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A solid line indicates that points on the line are included in the solution set (≥ or ≤), while a dashed line indicates that points on the line are not included (> or <).

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How do 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 x and y values of the point into the inequality. If the inequality holds true, then the point is a solution.

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What does shading represent in the graph of a linear inequality?

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Shading represents the set of all solutions to the inequality. The shaded area indicates where the inequality is true.

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What is the significance of the boundary line in a system of inequalities?

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The boundary line separates the solution region from the non-solution region. It helps to visually identify where the solutions to the inequalities lie.

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What does it mean if two lines in a system of inequalities intersect?

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If two lines intersect, it means there is a point that satisfies both inequalities, which can be a potential solution to the system.

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How can you tell if a system of inequalities has no solution?

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A system has no solution if the lines representing the inequalities are parallel and do not intersect, meaning there is no point that satisfies both inequalities.

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What is the solution set of a system of linear inequalities?

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The solution set is the collection of all points that satisfy all inequalities in the system, often represented as a shaded region on a graph.

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What does the term 'feasible region' refer to in linear inequalities?

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The feasible region is the area on a graph where all the inequalities in a system are satisfied. It is the intersection of all the shaded areas.