Rational equation

Definition

A rational equation is an equation that involves at least one rational expression, which is a fraction with a polynomial in both the numerator and the denominator. Solving these equations typically involves finding a common denominator or multiplying through by the least common denominator to clear fractions.

5 Must Know Facts For Your Next Test

1. To solve a rational equation, first find a common denominator for all terms involved.
2. Multiplying through by the least common denominator can help eliminate fractions and simplify the equation.
3. Always check for extraneous solutions, as some potential solutions may not satisfy the original equation after substituting back.
4. Cross-multiplication can be used when dealing with simple rational equations of the form $\frac{a}{b} = \frac{c}{d}$.
5. Rational equations can have no solution, one solution, or multiple solutions depending on the nature of the polynomials involved.

Review Questions

• What is a common method for eliminating fractions in a rational equation?
• Why is it important to check for extraneous solutions when solving rational equations?
• How do you approach solving an equation involving multiple rational expressions?

Related terms

Rational Expression: A fraction where both the numerator and the denominator are polynomials.

Least Common Denominator (LCD): The smallest expression that is a common multiple of the denominators of two or more fractions.

Extraneous Solution: A solution derived from an algebraic manipulation that does not satisfy the original equation.