Solving Rational Equations and Checking for Extraneous Roots Cheat Sheet

Key Facts

• A rational equation contains at least one rational expression, such as $\frac{x+2}{x-3}=5$.
• Domain restrictions come from denominator values that cannot be zero, so if $x-3 \neq 0$, then $x \neq 3$.
• The least common denominator, or LCD, is the smallest expression that every denominator divides evenly into.
• To clear denominators, multiply every term on both sides of the equation by the LCD.
• For a proportion $\frac{a}{b}=\frac{c}{d}$ with $b \neq 0$ and $d \neq 0$, cross multiplication gives $ad=bc$.
• After clearing denominators, solve the resulting linear, quadratic, or polynomial equation using normal algebra rules.
• A solution is extraneous if it makes any original denominator equal $0$ or fails when substituted into the original equation.
• The final answer set includes only values that satisfy the original rational equation and all domain restrictions.

Vocabulary

Rational Expression

A rational expression is a fraction whose numerator and denominator are polynomials, such as $\frac{x+1}{x^2-4}$.

Rational Equation

A rational equation is an equation that contains one or more rational expressions.

Denominator Restriction

A denominator restriction is a value that a variable cannot equal because it would make a denominator $0$.

Least Common Denominator

The least common denominator is the smallest expression that can be used to clear all fractions in a rational equation.

Extraneous Root

An extraneous root is a value found during solving that does not satisfy the original equation.

Cross Multiplication

Cross multiplication is the rule that $\frac{a}{b}=\frac{c}{d}$ implies $ad=bc$ when $b \neq 0$ and $d \neq 0$.