Solving Rational Equations

Front

What is a rational equation?

Back

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

Front

How do you solve a rational equation?

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

Front

What is a common denominator?

Back

A common denominator is a shared multiple of the denominators of two or more fractions, allowing them to be combined or compared.

Front

What does it mean to check for extraneous solutions?

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Checking for extraneous solutions means verifying that the solutions obtained from solving the equation do not make any denominator zero in the original equation.

Front

What is the first step in solving the equation \( \frac{1}{x-2} + \frac{2}{x+3} = 1 \)?

Back

The first step is to find a common denominator, which in this case is \( (x-2)(x+3) \).

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What is the importance of factoring in solving rational equations?

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Factoring helps simplify the equation and can reveal potential solutions or restrictions based on the factors.

Front

What is a restriction in a rational equation?

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A restriction is a value that cannot be included in the solution set because it would make a denominator zero.

Front

How do you find the least common denominator (LCD) of \( \frac{1}{x} \) and \( \frac{1}{x^2} \)?

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The least common denominator (LCD) is \( x^2 \) because it is the smallest expression that both denominators can divide into.

Front

What is the process of cross-multiplication in rational equations?

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Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction to eliminate the fractions.

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What is an example of a rational equation?

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An example of a rational equation is \( \frac{2}{x+1} = \frac{3}{x-2} \).