Solving 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 linear inequality?

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To graph a linear inequality, first graph the corresponding equation as a line. Use a solid line for ≤ or ≥ and a dashed line for < or >. Then shade the region that satisfies the inequality.

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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 shaded region of the graph.

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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 (≤ or ≥), while a dashed line indicates that points on the line are not included ( < or >).

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What is the solution set of a system of inequalities?

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The solution set is the collection of all points that satisfy all inequalities in the system, often represented as a shaded region on a graph.

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How can 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 does it mean if a system of inequalities has no solution?

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A system of inequalities has no solution if the shaded regions of the inequalities do not overlap, meaning there are no points that satisfy all inequalities.

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

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A system of inequalities has infinitely many solutions if the shaded regions overlap in such a way that there are an infinite number of points that satisfy all inequalities.

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What is the graphical representation of the inequality y > 2?

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The graphical representation of y > 2 is a dashed horizontal line at y = 2, with the region above the line shaded.

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How do you solve a system of inequalities by graphing?

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To solve by graphing, graph each inequality on the same coordinate plane, identify the overlapping shaded region, which represents the solution set.