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 that share 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 coordinates of the point 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 an infinite number of solutions?

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

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What is the graphical representation of a system of inequalities?

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The graphical representation consists of shaded regions on a coordinate plane, where each region corresponds to the solutions of the inequalities.

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

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The boundary line represents the points where the inequality changes from true to false. If the inequality is strict (e.g., > or <), the line is dashed; if it is inclusive (e.g., ≥ or ≤), the line is solid.

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How do you graph a linear inequality?

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1. Graph the boundary line. 2. Use a dashed line for strict inequalities and a solid line for inclusive inequalities. 3. Shade the region that satisfies the inequality.

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What does the solution set of a system of inequalities look like?

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The solution set is the intersection of the shaded regions of all inequalities, representing all points that satisfy every inequality in the system.

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What is the difference between a strict and non-strict inequality?

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A strict inequality (e.g., < or >) does not include the boundary line, while a non-strict inequality (e.g., ≤ or ≥) includes the boundary line.

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How can you tell if a point is above or below a line in a linear inequality?

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If the point satisfies the inequality (e.g., y > mx + b), it is above the line; if it does not satisfy it, it is below the line.

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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 where all the shaded regions overlap, representing all possible solutions to the system.