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 linear equation as a dashed or solid line, then shade the appropriate region based on the inequality symbol (e.g., above the line for 'greater than' and below for 'less than').

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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 either greater than or equal to a certain number.

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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 either less than or equal to a certain number.

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What is the solution set of a system of inequalities?

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The solution set is the region on the graph where the shaded areas of all inequalities overlap. It represents all possible solutions that satisfy all inequalities.

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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 difference between a dashed line and a solid line in graphing inequalities?

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A dashed line indicates that points on the line are not included in the solution (for '<' or '>'), while a solid line indicates that points on the line are included (for '≤' or '≥').

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What does it mean if a point lies outside the shaded region of a graph of inequalities?

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If a point lies outside the shaded region, it does not satisfy at least one of the inequalities in the system, meaning it is not a solution.

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How do you solve a system of inequalities algebraically?

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To solve algebraically, isolate one variable in one inequality, then substitute it into the other inequalities to find the range of values that satisfy all conditions.

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What is a feasible region in the context of systems of inequalities?

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The feasible region is the area on the graph that represents all possible solutions to the system of inequalities.