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 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 graph a system of inequalities?

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1. Graph each inequality as if it were an equation (using a solid line for ≤ or ≥ and a dashed line for < or >). 2. Shade the region that satisfies the inequality. 3. The solution to the system is where the shaded regions overlap.

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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 overlapping shaded region when graphed.

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

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The first step is to isolate one variable in one of the inequalities, if possible, to make graphing easier.

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How can you determine if a point is a solution to a system of inequalities?

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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 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 of the inequalities 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 a way that forms a continuous area, allowing for an infinite number of points.

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

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Identify the variables, translate the conditions into inequalities, and write them as a system that represents the relationships described in the problem.

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

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The boundary line separates the solution region from the non-solution region and helps determine which side of the line satisfies the inequality.