Solving 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 < or > and a solid line for ≤ or ≥. Then, shade the region that satisfies the inequality.

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What does it mean for a point to be a solution to a system of inequalities?

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A point is a solution to a system of inequalities if it satisfies all inequalities in the system, meaning it lies in the shaded region of the graph.

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

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How do you rewrite an inequality in 'y=' form?

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To rewrite an inequality in 'y=' form, isolate y on one side of the inequality. For example, from 3x - 6y > 12, you would rearrange to get y < (1/2)x - 2.

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What is the solution set of a system of inequalities?

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The solution set of a system of inequalities is the region on the graph where the shaded areas of all inequalities overlap.

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What does it mean if a system of inequalities has no solution?

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A system of inequalities has no solution if the shaded regions do not overlap, meaning there are no points that satisfy all inequalities.

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What does it mean if a system of inequalities has infinitely many solutions?

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A system of inequalities has infinitely many solutions if the shaded regions overlap in such a way that there are an infinite number of points that satisfy all inequalities.

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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 a solution.

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

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The boundary line represents the points where the inequalities change from true to false. It helps to define the regions that are solutions.