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 ordered pairs 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. Use a dashed line for inequalities that do not include equality (e.g., < or >) and a solid line for those that do (e.g., ≤ or ≥). Then, shade the region that satisfies each 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 the inequalities in the system. This means that when the coordinates of the point are substituted into each inequality, the resulting statements are true.

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What is the significance of 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 overlap. It represents all possible solutions to the system of inequalities.

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What is an ordered pair?

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An ordered pair is a pair of numbers used to represent a point in a coordinate plane, written in the form (x, y), where x is the horizontal coordinate and y is the vertical coordinate.

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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 x and y values into each inequality. If the ordered pair makes all inequalities true, then it is a solution.

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

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A system of inequalities has no solutions if there is no ordered pair that satisfies all inequalities simultaneously. This often occurs when the feasible regions do not overlap.

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What does it mean if a system of inequalities has infinite solutions?

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A system of inequalities has infinite solutions if there are countless ordered pairs that satisfy all inequalities. This typically happens when the inequalities represent the same line or overlapping regions.

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

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The boundary line separates the coordinate plane into two regions: one that satisfies the inequality and one that does not. The line itself may or may not be included in the solution set, depending on whether the inequality is strict or inclusive.

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How can you identify the maximum or minimum values in a system of inequalities?

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To identify maximum or minimum values, graph the system and look for the highest or lowest points in the feasible region, often at the vertices of the region.