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 ordered pairs that satisfy all inequalities in the system.

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How do you determine if an ordered pair is a solution to a system of inequalities?

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To determine if an ordered pair is a solution, substitute the values of the ordered pair into each inequality. If the pair satisfies all inequalities, it is a solution.

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What does the graph of a linear inequality represent?

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The graph of a linear inequality represents all the solutions to that inequality, typically shown as a shaded region on one side of the boundary line.

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What is the difference between '>' and '≥' in inequalities?

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'>' means that the value is strictly greater than, while '≥' means that the value is greater than or equal to.

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What is the significance of the boundary line in a linear inequality?

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The boundary line represents the values that make the inequality an equation. If the inequality is strict ('>' or '<'), the line is dashed; if it includes equality ('≥' or '≤'), the line is solid.

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How do you graph a system of inequalities?

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1. Graph each inequality as if it were an equation. 2. Use a dashed line for strict inequalities and a solid line for inclusive inequalities. 3. Shade the appropriate region for each inequality. 4. The solution is where the shaded regions overlap.

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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 lies in the overlapping shaded region of the graph of the inequalities.

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How can you check if a point is a solution to a system of inequalities?

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

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What is the solution set of a system of inequalities?

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The solution set is the set of all ordered pairs that satisfy all inequalities in the system, often represented graphically.

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What does the term 'feasible region' refer to 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.