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 determine if a point is a solution to a system of linear 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 part of the solution set.

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What does the graph of a linear inequality represent?

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The graph of a linear inequality represents all the solutions to the inequality, typically shown as a shaded region on one side of the boundary line.

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What is the difference between a solid line and a dashed line in the graph of linear 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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What is the feasible region in a system 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, representing all possible solutions.

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How can you find the intersection of two linear inequalities?

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To find the intersection, graph both inequalities and identify the region where they overlap. This region represents the solutions to both inequalities.

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What is the significance of the vertices in the feasible region?

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The vertices of the feasible region are important because they are potential solutions to optimization problems, often yielding maximum or minimum values.

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What is a constraint in the context of linear inequalities?

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A constraint is a condition that must be satisfied in a linear inequality, limiting the possible values of the variables.

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How do you write a system of linear inequalities from a word problem?

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Identify the variables, translate the conditions into inequalities, and express them as a system of inequalities.

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What is the graphical method for solving a system of linear inequalities?

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The graphical method involves graphing each inequality on the same coordinate plane and identifying the feasible region.