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?

Back

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 any solutions found do not make the original equation undefined or false.

Front

What is an example of an extraneous solution?

Back

If solving \( \frac{1}{x} = 2 \) gives \( x = \frac{1}{2} \), substituting back shows it is valid. However, if \( x = 0 \) is found, it is extraneous because it makes the original equation undefined.

Front

What is the importance of factoring in solving rational equations?

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Factoring helps simplify the equation, making it easier to find solutions by breaking down polynomials into their roots.

Front

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

Back

The least common denominator (LCD) is \( x(x-2) \), which is the product of the unique factors of the denominators.

Front

What is a vertical asymptote in the context of rational functions?

Back

A vertical asymptote is a line \( x = a \) where the function approaches infinity or negative infinity, typically where the denominator equals zero.

Front

What is the significance of the domain in rational equations?

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The domain of a rational equation is the set of all possible values of the variable that do not make the denominator zero.