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 that share the same variables. The solution to the system is the set of all points that satisfy all inequalities in the system.

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How do you represent a situation with inequalities?

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To represent a situation with inequalities, identify the constraints and conditions of the problem, then translate them into mathematical inequalities using variables.

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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 hours spent on two activities (x and y) cannot exceed 6 hours.

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What does the inequality 32x + 20y ≥ 150 signify in a baking scenario?

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This inequality signifies that the total number of cookies baked (32 chocolate chip cookies per hour and 20 snickerdoodle cookies per hour) must be at least 150.

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What is the graphical representation of a system of inequalities?

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The graphical representation of a system of inequalities involves shading the regions of the graph that satisfy each inequality, with the solution being the intersection of these shaded regions.

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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 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 + 2y ≥ 4 represent?

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This inequality represents a condition where the combination of x and y must meet or exceed a certain threshold (4) based on their coefficients.

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What is the significance of the inequality 3x - 2y < 4 in a system?

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This inequality indicates a constraint where the linear combination of x and y must be less than 4, affecting the feasible region of solutions.

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

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A solution satisfies all inequalities in the system, while a non-solution does not meet at least one of the inequalities.

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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 appropriate region based on the inequality.