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/or denominator are polynomials.

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How do you solve a rational inequality?

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To solve a rational inequality, first set the rational expression to zero, find the critical points, test intervals between these points, and determine where the inequality holds true.

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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 its endpoints a and b, while a closed interval [a, b] includes both endpoints.

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

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A rational inequality is undefined when the denominator of the rational expression equals zero, as division by zero is not allowed.

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What is the significance of critical points in solving rational inequalities?

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Critical points are values where the rational expression is either zero or undefined; they help to divide the number line into intervals for testing.

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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 the union of intervals in the context of rational inequalities?

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The union of intervals combines multiple intervals into a single solution set, indicating that any value within any of the intervals satisfies the inequality.

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What is the role of test points in solving rational inequalities?

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Test points are specific values chosen from each interval created by critical points to determine whether the inequality holds true in that interval.

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How do you determine if a rational inequality is true for a specific interval?

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To determine if a rational inequality is true for a specific interval, substitute a test point from that interval into the inequality and check if the resulting statement is true.

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What is the graphical representation of a rational inequality?

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The graphical representation of a rational inequality shows the number line with shaded regions indicating the solution set, often marked with open or closed circles at critical points.