Rational Equation Calculator

Solve rational equations step-by-step. Enter equations with fractions and variables, get detailed solutions.

Format: numerator/denominator = value

Solution

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Domain Restrictions

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Equation Summary

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Step-by-Step Solution

Enter a rational equation and click "Solve Equation" to see the step-by-step solution.

Solving Method

  1. Find LCD (Least Common Denominator)
  2. Multiply both sides by LCD
  3. Simplify the equation
  4. Solve for the variable
  5. Check domain restrictions
  6. Verify solution

Domain Restrictions

  • Denominator ≠ 0: Values that make denominator zero are excluded
  • Check solutions: Verify solutions don't violate domain
  • Extraneous solutions: Solutions that don't work in original equation

Common Examples

  • 1/x = 2: x = 1/2
  • 1/(x+1) = 2: x = -1/2
  • 1/(x+1) + 1/(x-2) = 3: x = (1 ± √37)/6
  • 3/(x-2) = 5: x = 13/5

Frequently Asked Questions

What is a rational equation?

A rational equation is an equation containing at least one fraction whose numerator and denominator are polynomials. Examples include 1/x = 2, (x+1)/(x-2) = 3, or 1/(x+1) + 1/(x-2) = 4.

How do you solve rational equations?

  1. Find the LCD: Determine the least common denominator of all fractions
  2. Multiply through: Multiply both sides of the equation by the LCD
  3. Simplify: Cancel denominators and simplify the resulting equation
  4. Solve: Solve the resulting polynomial equation
  5. Check: Verify solutions don't make any denominator zero

What are extraneous solutions?

Extraneous solutions are values that appear to satisfy the transformed equation but don't work in the original rational equation. They occur when multiplying by variables that could be zero. Always check your solutions in the original equation!

How do you solve 1/(x+1) = 2?

Multiply both sides by (x+1): 1 = 2(x+1) → 1 = 2x + 2 → 2x = -1 → x = -1/2. Check: 1/(-1/2+1) = 1/(1/2) = 2 ✓

How do you solve 1/x + 1/(x+1) = 1?

LCD = x(x+1). Multiply: (x+1) + x = x(x+1) → 2x+1 = x²+x → x² - x - 1 = 0 → x = (1 ± √5)/2. Check both solutions in original equation.

Why does my rational equation have no solution?

A rational equation may have no solution if:

  • All possible solutions are extraneous (make denominator zero)
  • The equation simplifies to a contradiction (e.g., 0 = 1)
  • The solution requires division by zero

What is the LCD and why do I need it?

The Least Common Denominator (LCD) is the smallest expression divisible by all denominators in the equation. Multiplying by the LCD eliminates all fractions, making the equation easier to solve. The LCD is essential for solving rational equations.

Can I just cross-multiply?

Cross-multiplication works for equations with a single fraction on each side (a/b = c/d). For complex equations with multiple terms (a/b + c/d = e), you need to use the LCD method. This calculator handles both cases.