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 is the set of all points that satisfy all inequalities in the system.

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How do you graph a system of inequalities?

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To graph a system of inequalities, first graph each inequality as if it were an equation. Use a dashed line for inequalities that do not include equality (e.g., < or >) and a solid line for those that do (e.g., ≤ or ≥). Then, shade the appropriate region for each inequality, and the solution set is where the shaded regions overlap.

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What does the solution set of a system of inequalities represent?

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The solution set of a system of inequalities represents all the points (x, y) that satisfy all the inequalities in the system.

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What is the difference between a solid line and a dashed line in 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 >).

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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 x and y values of the point into each inequality. If the point satisfies all inequalities, it is part of the solution set.

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What does the inequality y < 2x - 4 represent graphically?

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The inequality y < 2x - 4 represents the region below the line y = 2x - 4, where the line itself is not included in the solution set.

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What is the significance of the intersection of shaded regions in a graph of inequalities?

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The intersection of shaded regions represents the solution set of the system of inequalities, where all conditions are satisfied.

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How do you write a system of inequalities from a word problem?

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To write a system of inequalities from a word problem, identify the variables, determine the relationships described in the problem, and express these relationships as inequalities.

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What is the role of the boundary line in a system of inequalities?

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The boundary line separates the solution set from the non-solution set. It helps to visually determine which side of the line satisfies the inequality.

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What does the point (4, 3) indicate in the context of a system of inequalities?

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The point (4, 3) indicates a specific solution that satisfies all inequalities in the system.