Systems of Inequalities Exam Review

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Define 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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What does it mean to graph a system of inequalities?

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Graphing a system of inequalities involves plotting each inequality on the same coordinate plane and identifying the region where all inequalities overlap, which represents the solution set.

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Identify the variables in a ticket sales problem.

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In a ticket sales problem, variables typically represent the quantities of tickets sold in different categories, such as 'x' for tickets sold at the door and 'y' for tickets sold in advance.

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What is the significance of the vertices in a system of inequalities?

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Vertices are the corner points of the feasible region formed by the intersection of the inequalities. They are critical for finding the maximum or minimum values of a linear objective function.

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How do you write an inequality for a maximum constraint?

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A maximum constraint is written as an inequality with a less than or equal to symbol (≤), indicating that the variable cannot exceed a certain value.

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How do you write an inequality for a minimum constraint?

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A minimum constraint is written as an inequality with a greater than or equal to symbol (≥), indicating that the variable must be at least a certain value.

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What is the objective function in a profit maximization problem?

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The objective function is a mathematical expression that represents the profit or cost to be maximized or minimized, typically in terms of the variables defined in the problem.

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How do you determine the feasible region in a system of inequalities?

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The feasible region is determined by graphing all inequalities and identifying the area where all shaded regions overlap, representing all possible solutions.

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What is the role of constraints in a system of inequalities?

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Constraints limit the values that the variables can take, defining the boundaries of the feasible region and ensuring that solutions meet specific conditions.

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Explain the concept of linear programming.

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Linear programming is a method used to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships, often used for maximizing profit or minimizing costs.