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 system of inequalities?

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To graph a system of inequalities, first graph each inequality as if it were an equation. Then, shade the appropriate region for each inequality. The solution set is where the shaded regions overlap.

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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 can be less than or equal to a certain number.

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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 can be greater than or equal to a certain number.

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What is the difference between a solution to an equation and a solution to an inequality?

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A solution to an equation is a specific value that makes the equation true, while a solution to an inequality is a range of values that satisfy the inequality.

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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 is the feasible region in a system of inequalities?

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The feasible region is the area on the graph where all the shaded regions of the inequalities overlap. It represents all possible solutions.

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

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If a system of inequalities has no solution, it means that there is no set of values that satisfies all inequalities simultaneously.

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What is the significance of the boundary line in a graph of an inequality?

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The boundary line represents the values where the inequality changes from true to false. If the inequality is strict (>, <), the line is dashed; if it includes equality (≥, ≤), the line is solid.

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How do you write an inequality from a word problem?

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Identify the quantities involved, determine the relationship (greater than, less than, etc.), and express it using inequality symbols.