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 represent a constraint in a system of inequalities?

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Constraints in a system of inequalities are represented using inequality symbols (>, <, ≥, ≤) to show the limits on the variables.

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What does the solution set of a system of inequalities look like?

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The solution set of a system of inequalities is often represented as a shaded region on a graph, indicating all possible solutions that satisfy the inequalities.

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What is the difference between a system of equations and a system of inequalities?

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Systems of equations have specific solutions (one, none, or infinitely many), while systems of inequalities can have a range of solutions.

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How do you graph a linear inequality?

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

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What does the symbol '≥' mean in an inequality?

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The symbol '≥' means 'greater than or equal to', indicating that the value can be either greater than or equal to a certain number.

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What does the symbol '≤' mean in an inequality?

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The symbol '≤' means 'less than or equal to', indicating that the value can be either less than or equal to a certain number.

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If x + y ≤ 6, what does this represent in a real-world context?

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This represents a constraint where the total number of items (x and y) cannot exceed 6, such as the maximum number of hours available for a task.

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What is the significance of the shaded region in a graph of inequalities?

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The shaded region represents all possible solutions that satisfy the inequality, showing where the values of the variables can exist.

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How can 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 point's coordinates into each inequality. If the point satisfies all inequalities, it is a solution.