Problem Solving: Systems of Linear Inequalities

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What is a system of linear inequalities?

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A system of linear inequalities is a set of two or more linear inequalities that involve 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 represent a budget constraint in a linear inequality?

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A budget constraint can be represented as an inequality where the total cost of items does not exceed the budget. For example, if x is the number of item A and y is the number of item B, the inequality can be written as: cost_A * x + cost_B * y <= budget.

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What does the symbol '≥' represent in inequalities?

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The symbol '≥' means 'greater than or equal to'. It indicates that the value on the left side of the inequality is either greater than or equal to the value on the right side.

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What does the symbol '≤' represent in inequalities?

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The symbol '≤' means 'less than or equal to'. It indicates that the value on the left side of the inequality is either less than or equal to the value on the right side.

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How do you write an inequality for a minimum requirement?

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An inequality for a minimum requirement is written using the '≥' symbol. For example, if a minimum of 8 people is required, it can be expressed as: x + y ≥ 8.

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How do you write an inequality for a maximum capacity?

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An inequality for a maximum capacity is written using the '≤' symbol. For example, if the maximum capacity is 455 people, it can be expressed as: x ≤ 455.

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What is the first step in solving a system of linear inequalities?

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The first step in solving a system of linear inequalities is to graph each inequality on the same coordinate plane to find the feasible region where all inequalities overlap.

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What does the feasible region represent in a system of linear inequalities?

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The feasible region represents all possible solutions that satisfy all inequalities in the system. It is typically shaded on a graph.

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How do you determine if a point is a solution to a system of linear 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 significance of the boundary line in a linear inequality?

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The boundary line represents the points where the inequality changes from true to false. If the inequality is '≥' or '≤', the line is included in the solution (solid line). If it is '>' or '<', the line is not included (dashed line).