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 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 for a point to be a solution to a system of inequalities?

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A point is a solution if it lies in the region defined by the inequalities, meaning it satisfies all the inequalities in the system.

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What is the graphical representation of a system of inequalities?

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The graphical representation consists of shaded regions on a coordinate plane, where each region corresponds to the solutions of the inequalities.

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What is the significance of the boundary line in a system of inequalities?

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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 graph the inequality y > 2x - 4?

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1. Graph the line y = 2x - 4 as a dashed line. 2. Shade the region above the line, as y must be greater than the line.

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What does the solution set of a system of inequalities represent?

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The solution set represents all the points that satisfy all inequalities in the system, often visualized as a shaded region on a graph.

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What is the difference between a solution and a non-solution in a system of inequalities?

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A solution satisfies all inequalities, while a non-solution does not satisfy at least one inequality.

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How can you identify a point that is not part of the solution set?

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Substitute the point's coordinates into each inequality. If it fails to satisfy any inequality, it is not part of the solution set.

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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 regions of the graph satisfy the inequalities. By testing points in different regions, you can identify the solution set.