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 region that satisfies the inequality. The solution to the system 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 satisfies all inequalities in the system when substituted into them.

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What is the difference between a solid line and a dashed line in graphing inequalities?

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

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

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What is the significance of the vertices of the feasible region?

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The vertices of the feasible region are important because they are potential solutions to the system of inequalities and can be used to find optimal solutions in linear programming.

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How do you find the intersection of two inequalities?

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To find the intersection of two inequalities, graph both inequalities and identify the region where their shaded areas overlap.

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What is a boundary line in the context of inequalities?

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A boundary line is the line that represents the equation of the inequality. It divides the coordinate plane into two half-planes.

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What does it mean for a system of inequalities to be inconsistent?

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A system of inequalities is inconsistent if there is no solution that satisfies all inequalities, meaning the feasible region is empty.