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

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What does the symbol '≥' mean in an inequality?

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'≥' means 'greater than or equal to'. It indicates that the value on the left is either greater than or equal to the value on the right.

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What does the symbol '>' mean in an inequality?

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'>' means 'greater than'. It indicates that the value on the left is strictly greater than the value on the right.

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What does the symbol '≤' mean in an inequality?

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'≤' means 'less than or equal to'. It indicates that the value on the left is either less than or equal to the value on the right.

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What does the symbol '<' mean in an inequality?

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'<' means 'less than'. It indicates that the value on the left is strictly less than the value on the right.

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How can 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. Use a solid line for '≥' or '≤' and a dashed line for '>' or '<'. Then shade the appropriate region that satisfies each inequality.

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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 shaded regions overlap. It represents all possible solutions to the system of inequalities.

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What is an example of a real-world application of systems of inequalities?

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An example is determining the number of adult and child tickets sold at a theater, where the total number of tickets cannot exceed the seating capacity and the revenue must meet a certain amount.

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

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Boundary lines represent the limits of the inequalities. Points on the line satisfy the equation, while the shaded area indicates solutions that satisfy the inequality.