Solving Rational Equations

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What is a rational equation?

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A rational equation is an equation that involves at least one rational expression, which is a fraction where the numerator and denominator are polynomials.

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How do you solve a rational equation?

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To solve a rational equation, find a common denominator, eliminate the fractions by multiplying both sides by that denominator, and then solve the resulting polynomial equation.

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What are extraneous solutions?

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Extraneous solutions are solutions that emerge from the process of solving an equation but do not satisfy the original equation.

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Why is it important to check for extraneous solutions in rational equations?

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It is important to check for extraneous solutions because they can lead to incorrect answers that do not satisfy the original equation.

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What is the first step in solving a rational equation?

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The first step is to identify the least common denominator (LCD) of all the rational expressions in the equation.

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What happens if you multiply both sides of a rational equation by the denominator?

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Multiplying both sides by the denominator eliminates the fractions, but it can also introduce extraneous solutions.

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How can you verify if a solution is valid for a rational equation?

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To verify if a solution is valid, substitute it back into the original equation and check if both sides are equal.

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What is the least common denominator (LCD)?

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The least common denominator (LCD) is the smallest expression that can be used as a common denominator for two or more fractions.

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What is a polynomial?

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A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents.

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What is the difference between a rational expression and a rational equation?

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A rational expression is a fraction involving polynomials, while a rational equation is an equation that contains at least one rational expression.