Rational Inequalities

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What is a rational inequality?

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A rational inequality is an inequality that involves a rational expression, which is a fraction where the numerator and denominator are polynomials.

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How do you solve a rational inequality?

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To solve a rational inequality, first find the critical points by setting the numerator and denominator to zero. Then, test intervals between these points to determine where the inequality holds true.

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What does it mean for a rational inequality to be undefined?

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A rational inequality is undefined at points where the denominator equals zero, as division by zero is not possible.

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What is the significance of critical points in rational inequalities?

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Critical points are the values where the rational expression equals zero or is undefined. They help in determining the intervals to test for the solution of the inequality.

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What is the difference between open and closed intervals?

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An open interval (a, b) does not include the endpoints a and b, while a closed interval [a, b] includes both endpoints.

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How do you represent the solution set of a rational inequality?

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The solution set of a rational inequality can be represented using interval notation, which indicates the ranges of values that satisfy the inequality.

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What is interval notation?

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Interval notation is a mathematical notation used to represent a range of values. For example, (1, 5) represents all numbers between 1 and 5, not including 1 and 5.

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What does the symbol '∪' represent in interval notation?

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The symbol '∪' represents the union of two sets, meaning that it combines the elements of both sets.

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How do you determine the sign of a rational expression in an interval?

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To determine the sign of a rational expression in an interval, choose a test point from the interval and substitute it into the expression. The sign of the result indicates whether the expression is positive or negative in that interval.

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What is the importance of testing intervals in solving rational inequalities?

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Testing intervals helps to identify where the rational expression is positive or negative, which is essential for determining the solution set of the inequality.