System 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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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 coordinates of the point into each inequality. If the point satisfies all inequalities, it is a solution.

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What does it mean if a system of linear inequalities has no solution?

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A system has no solution if there is no point that satisfies all inequalities, often represented by parallel lines that do not intersect.

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What does it mean if a system of linear inequalities has infinitely many solutions?

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A system has infinitely many solutions if the inequalities overlap in such a way that there are countless points that satisfy all inequalities, often represented by overlapping lines.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a region of the coordinate plane that is either shaded above or below a line, depending on the inequality sign.

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What is the difference between a strict inequality and a non-strict inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., ≤ or ≥) includes the boundary line.

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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 dashed line (for strict inequalities) or solid line (for non-strict inequalities), then shade the appropriate region.

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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. Points on the line may or may not be included in the solution, depending on the type of inequality.

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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 where all the constraints (inequalities) of a linear programming problem are satisfied.

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What is the role of test points in solving systems of inequalities?

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Test points are used to determine which side of the boundary line is part of the solution set by substituting the test point into the inequalities.