Graphing 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 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 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 such a way that there are an infinite number of points that satisfy all inequalities.

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What is the graphical representation of a linear inequality?

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The graphical representation of a linear inequality is a half-plane, which is the area above or below the line, depending on the inequality sign.

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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 graph a system of inequalities?

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To graph a system of inequalities, graph each inequality on the same coordinate plane, shade the appropriate region for each, and identify the overlapping shaded area.

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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 to the inequality. Any point within this region satisfies the inequality.

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What does it mean for two inequalities to be parallel?

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Two inequalities are parallel if their corresponding lines have the same slope but different y-intercepts, meaning they will never intersect.

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Can a system of parallel inequalities have a solution?

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Yes, a system of parallel inequalities can have a solution if the inequalities are not strict (i.e., they include equalities), allowing for overlapping shaded regions.