Solving/interpreting linear inequalities in two variables

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What is a linear inequality in two variables?

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A linear inequality in two variables is an inequality that can be expressed in the form ax + by <, >, ≤, or ≥ c, where a, b, and c are constants and x and y are variables.

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How do you graph a linear inequality in two variables?

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To graph a linear inequality, first graph the corresponding linear equation as a boundary line. Then, use a dashed line for < or > and a solid line for ≤ or ≥. Finally, shade the region that satisfies the inequality.

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What is the objective function in a linear programming problem?

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The objective function is a mathematical expression that defines the goal of the optimization problem, typically to maximize or minimize a quantity, such as profit or cost.

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What does it mean to maximize profit in a linear programming context?

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Maximizing profit means finding the combination of variables (e.g., products) that results in the highest possible profit, given certain constraints.

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What are constraints in a linear programming problem?

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Constraints are the limitations or restrictions on the variables in a linear programming problem, often expressed as linear inequalities.

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How do you write a system of inequalities?

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A system of inequalities consists of two or more inequalities that represent the constraints of a linear programming problem, and they must be solved simultaneously.

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What is the significance of the feasible region in linear programming?

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The feasible region is the area on a graph that represents all possible solutions that satisfy the constraints of a linear programming problem.

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How do you determine the corner points of the feasible region?

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Corner points of the feasible region can be found by identifying the intersection points of the boundary lines of the inequalities.

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What is the graphical method for solving linear programming problems?

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The graphical method involves graphing the constraints, identifying the feasible region, and evaluating the objective function at the corner points to find the optimal solution.

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

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