A.EI.2f Solutions to 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 linear inequality?

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To graph a linear inequality, first graph the corresponding linear equation as a solid line (for ≤ or ≥) or a dashed line (for < or >). 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 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 (≥ or ≤), while a dashed line indicates that points on the line are not included (> or <).

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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 is the feasible region in the context of linear inequalities?

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The feasible region is the area on the graph where all the shaded regions of the inequalities overlap. It represents all possible solutions to the system.

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What does it mean if a point lies outside the feasible region?

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If a point lies outside the feasible region, it does not satisfy at least one of the inequalities in the system and is therefore not a solution.

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What is the significance of the intersection of two inequalities?

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The intersection of two inequalities represents the set of points that satisfy both inequalities simultaneously, forming part of the feasible region.

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Can a system of linear inequalities have no solution?

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Yes, a system of linear inequalities can have no solution if the inequalities represent parallel lines that do not intersect.

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What is an example of a system of linear inequalities?

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An example is: 2x + y < 4 and x - y ≥ 1. The solution is the set of points that satisfy both inequalities.