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 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 the inequality x + y ≤ 6 represent in a real-world context?

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This inequality represents a constraint where the total number of items x and y cannot exceed 6, such as hours spent on two activities.

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What is the significance of the inequality 32x + 20y ≥ 150 in a baking scenario?

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This inequality indicates that the total number of cookies baked (32 cookies per hour for x hours and 20 cookies per hour for y hours) must be at least 150.

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What does the term 'feasible region' mean in the context of systems of inequalities?

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The feasible region is the area on a graph where all the inequalities in the system overlap, representing all possible solutions.

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How can 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 appropriate region that satisfies the inequality.

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What is the difference between a solution and a feasible solution in a system of inequalities?

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A solution is any point that satisfies all inequalities, while a feasible solution is a solution that also meets all constraints of the problem.

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What does it mean if a point lies on the boundary line of an inequality?

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If a point lies on the boundary line, it satisfies the equation of the line but may or may not satisfy the inequality, depending on whether the inequality is strict or inclusive.

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What is the role of the maximum hours constraint in a real-world problem involving inequalities?

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The maximum hours constraint limits the total time available for activities, helping to define the feasible region for solutions.

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How do you interpret the solution (1, 0) in a system of inequalities?

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The solution (1, 0) means that one unit of activity x and zero units of activity y satisfy all inequalities in the system.