Rational equation

High School

Definition

An equation that contains at least one rational expression on one or both sides of the equation. The rational expresion is a fraction where the denominator and numerator are polynomials. Rational equations can be solved by reducing the common denominator and then solve for the numerator.

Worked examples

\(\frac{3}{x} + \frac{2}{x} = 5\)

Common denominator is \(x\); combine to get \(\frac{5}{x} = 5\), then \(5 = 5x\), so \(x = 1\).

\(\frac{1}{x-2} = \frac{3}{x+1}\)

Cross-multiply: \(1(x+1) = 3(x-2)\) gives \(x+1 = 3x-6\), so \(x = \frac{7}{2}\).

\(\frac{x}{x-3} - \frac{2}{x-3} = 1\)

Same denominator \(x-3\); combine numerators to get \(\frac{x-2}{x-3} = 1\), then \(x-2 = x-3\) has no solution.

Common mistakes

\(\frac{1}{x} + \frac{1}{x} = \frac{2}{x^2}\) → \(\frac{1}{x} + \frac{1}{x} = \frac{2}{x}\) Add numerators when denominators match; don't multiply denominators.
Forgetting to check if a solution makes a denominator zero → Always exclude values that make any denominator zero A solution that creates division by zero is extraneous and must be rejected.
\(\frac{x+3}{x} = 3 \)→\( x+3 = 3\) → \(\frac{x+3}{x} = 3 \)→\( x+3 = 3x\) Multiply both sides by the denominator \(x\), not just the right side.

Where you'll use it next

Rational equations appear in work and mixture problems, in simplifying complex fractions, and when solving rational functions. You'll use them extensively in precalculus, calculus (related rates, optimization), and physics (lens equations, electrical circuits).

Found in 2 StudyPug lessons

Solving Rational Equations

Grade 12 Math

Clear the fractions with the LCD, solve, then check the result against the domain restrictions.

12th Grade12th

Applications of Equations with Algebraic Fractions

Grade 11 Math

Turn work-rate, distance, and mixture word problems into rational equations you can solve step by step, with clear worked examples.

11th Grade11th