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 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 inequalities?

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To graph a system of inequalities, first graph each inequality as if it were an equation. Then, shade the region that satisfies the inequality. The solution to the system 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 an inequality?

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The boundary line represents the points where the inequality changes from true to false. If the inequality is strict (e.g., < or >), the line is dashed; if it is inclusive (e.g., ≤ or ≥), the line is solid.

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How do 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 a system of inequalities?

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

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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 write a system of inequalities from a word problem?

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Identify the constraints and objectives in the problem, define variables for the quantities involved, and translate the constraints into inequalities.

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What does it mean for a solution to be valid in a real-world context?

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A valid solution must not only satisfy the mathematical inequalities but also make sense in the context of the problem, such as not having negative quantities.

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What is the role of the objective function in optimization problems?

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The objective function is the equation that needs to be maximized or minimized, often subject to the constraints defined by the inequalities.