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

Back

The first step is to find a common denominator, which in this case is \( x(x+1) \).

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.

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What is an extraneous solution?

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An extraneous solution is a solution that emerges from the process of solving an equation but does not satisfy the original equation.

Front

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

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The least common denominator (LCD) of \( \frac{1}{x} \) and \( \frac{2}{x+1} \) is \( x(x+1) \).

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What is the importance of factoring in solving rational equations?

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Factoring is important because it simplifies the equation, making it easier to find solutions by setting each factor equal to zero.

Front

What is a polynomial?

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A polynomial is a mathematical expression that consists of variables, coefficients, and non-negative integer exponents, combined using addition, subtraction, and multiplication.

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 in an equation.