Systems of Linear Inequalities Practice

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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 set is the region where the graphs of the inequalities overlap.

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How do you determine if an ordered pair is a solution to a system of inequalities?

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To determine if an ordered pair is a solution, substitute the values of the ordered pair into each inequality. If the pair 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 the graphs of the inequalities do not intersect, meaning there is no ordered pair that satisfies all inequalities.

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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 graphs of the inequalities overlap in a region, allowing for an infinite number of ordered pairs to satisfy all inequalities.

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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 shaded region on one side of a boundary line, which represents all the solutions to the inequality.

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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 graph a system of linear inequalities?

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To graph a system of linear inequalities, graph each inequality as a line (solid or dashed) and shade the appropriate region. The solution set is where the shaded regions overlap.

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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 significance of the vertices of the feasible region?

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The vertices of the feasible region are important because they represent potential solutions to optimization problems, such as maximizing or minimizing a function.

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What does it mean for an ordered pair to be in the solution set of a system of inequalities?

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An ordered pair is in the solution set if it satisfies all the inequalities in the system.