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Algebra / Solving Equations and Inequalities

Rational Equations

A rational equation has a variable in a denominator. The method is always the same: multiply every term by the least common denominator so the fractions disappear, solve what's left, then check that no answer makes a denominator zero.

Practice Problems
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Chapter 2 · Lesson 21 of 28Solving Equations and Inequalities
  1. 1.Solving Equations and Inequalities
  2. 2.Solutions and Solution Sets
  3. 3.Linear Equations
  4. 4.Applications of Linear Equations
  5. 5.Equations with More Than One Variable
  6. 6.Linear Equations with Fractions
  7. 7.Word Problems
  8. 8.Distance, Rate, and Time Problems
  9. 9.Work Problems
  10. 10.Mixture Problems
  11. 11.Direct Variation
  12. 12.Inverse Variation
  13. 13.Joint Variation
  14. 14.Solving for a Variable
  15. 15.Quadratic Equations
  16. 16.Completing the Square
  17. 17.The Quadratic Formula
  18. 18.Applications of Quadratic Equations
  19. 19.Equations Reducible to Quadratic Form
  20. 20.Equations with Radicals
  21. 21.Rational Equations
  22. 22.Linear Inequalities
  23. 23.Compound Inequalities
  24. 24.Interval Notation
  25. 25.Polynomial Inequalities
  26. 26.Rational Inequalities
  27. 27.Absolute Value Equations
  28. 28.Absolute Value Inequalities
See all algebra chapters →

Fractions with a variable in the bottom look intimidating, but there’s a reliable way to deal with them: get rid of them. A rational equation is an equation containing rational expressions, fractions with polynomials in the numerator and denominator. Multiply everything by the right expression and the fractions vanish, leaving a linear or quadratic equation you already know how to solve. The one catch, and it’s an important one, is that some answers have to be thrown away at the end.

A rational equation P(x)/Q(x) = R(x)/S(x) becomes P(x) S(x) = R(x) Q(x) after clearing denominators, with Q(x) and S(x) not zero
Clear the fractions with the LCD, then solve

How to Solve Rational Equations

  1. Note the restrictions. Write down every value of that makes a denominator zero. Those can never be solutions.
  2. Find the LCD of all the denominators. Factor them first if they’re polynomials.
  3. Multiply every term, on both sides, by the LCD. Each fraction cancels to a polynomial.
  4. Solve the resulting equation.
  5. Check each answer against the restrictions, and reject any that match. Those are extraneous solutions.

This is the same idea as clearing fractions in linear equations with fractions. The only new part is that the denominators can now contain , which is where the restrictions come from.

The equation 3/x + 1/2 = 2 multiplied term by term by the LCD 2x: 2x times 3/x is 6, 2x times 1/2 is x, and 2x times 2 is 4x, giving 6 + x = 4x and x = 2
Multiplying every term by the LCD cancels every denominator

Example 1: Numbers in the Denominators

Solve .

Step 1 — the LCD of and is . Multiply every term by :

Step 2 — solve:

No variable is in a denominator, so there are no restrictions. Check: ✓.

Example 2: A Variable in the Denominator

Solve .

Step 1 — restriction: .

Step 2 — multiply every term by the LCD, :

Step 3 — solve:

Step 4 — check: , and ✓.

Example 3: A Proportion (Cross-Multiplying)

Solve .

When the equation is exactly one fraction equal to another, cross-multiplying is the quick route. It’s the LCD method in disguise.

Step 1 — restrictions: and .

Step 2 — cross-multiply:

Step 3 — check: ✓.

Example 4: Polynomial Denominators

Solve .

Step 1 — factor the denominators to find the LCD: , so the LCD is and the restrictions are and .

Step 2 — multiply every term by the LCD:

Step 3 — solve:

Step 4 — check: isn’t or , and both sides equal ✓.

Extraneous Solutions

When the Only Answer Is Rejected

Solve .

Step 1 — restriction: .

Step 2 — multiply every term by :

Step 3 — check: is exactly the restricted value. Putting it into the original equation divides by zero, so it’s extraneous.

The equation has no solution. The graph shows why: the two sides are the same curve shifted up by , so they never meet. The value is where both curves blow up, not where they cross.

The graphs of x/(x − 3) and 3/(x − 3) + 2 are identical hyperbolas shifted vertically by 1, with a dashed asymptote at x = 3; the curves never intersect, so the equation has no solution
The two sides never intersect; x = 3 is the asymptote, not a solution

When One of Two Answers Is Rejected

Solve .

Step 1 — factor: . The LCD is , and the restrictions are and .

Step 2 — multiply every term by :

Step 3 — factor and solve:

Step 4 — check against the restrictions:

Checking the two roots: x = −2 gives denominators −3 and 6 and both sides equal 1/3, so it is kept; x = 1 makes x − 1 and x² − x zero, so it is extraneous
Every candidate answer has to pass the denominator check

The only solution is .

Rational Equations That Become Quadratic

Solve .

Step 1 — multiply by (restriction ):

Step 2 — factor:

Both pass the check: ✓ and ✓. When the cleared equation is quadratic, expect up to two answers and check each one. If it doesn’t factor, the quadratic formula works too.

A Rational Equation Word Problem

One pipe fills a tank in hours and another in hours. How long do they take together?

Step 1 — write the rates: together they fill of the tank per hour, and if the job takes hours, that combined rate is also :

Step 2 — multiply by the LCD, :

Together they fill the tank in 3 hours. The work problems lesson covers more problems like this.

Common Mistakes to Avoid

  • Multiplying only some terms by the LCD. Every term on both sides gets multiplied, including whole numbers like the in Example 2.
  • Skipping the check. An answer that makes a denominator zero must be rejected, even if your algebra was perfect.
  • Cross-multiplying when there are extra terms. isn’t a single fraction on each side, so cross-multiplying doesn’t apply.
  • Not factoring denominators first. The LCD of and is , not their product.
  • Losing the negative in a subtracted fraction. Put numerators in brackets when you multiply through.

Practice Problems

Work each one before opening the answer. To check an equation of your own, the rational equations calculator lists the restrictions, clears the fractions and flags extraneous solutions.

Problem 1. Solve .

Show answer

Step 1 — multiply every term by :

Answer:

Problem 2. Solve .

Show answer

Step 1 — restrictions: , .

Step 2 — cross-multiply:

Answer:

Problem 3. Solve .

Show answer

Step 1 — multiply every term by (restriction ):

Answer:

Problem 4. Solve .

Show answer

Step 1 — multiply by :

Step 2 — factor:

Answer: or , and both check

Problem 5. Solve .

Show answer

Step 1 — restriction: .

Step 2 — multiply by :

Step 3 — is the restricted value, so it’s extraneous.

Answer: no solution

Problem 6. One printer finishes a job in hours and another in hours. How long do they take working together?

Show answer

Step 1 — set up the equation:

Step 2 — multiply by :

Answer: hours

Rational equations ask where two rational expressions are equal. When the question is where one is bigger, the denominators matter even more, because their sign can flip an inequality. That’s the subject of rational inequalities.

Frequently Asked Questions

How do you solve rational equations?+

Find the least common denominator (LCD) of every fraction, multiply every term on both sides by it, and solve the resulting equation, which has no fractions. Then check each answer in the original equation and reject any that make a denominator equal to zero.

What is an extraneous solution of a rational equation?+

A value that solves the equation you got after clearing fractions but not the original equation, usually because it makes a denominator zero. For example, solving x/(x − 3) = 3/(x − 3) + 2 gives x = 3, but x = 3 makes the denominators zero, so it is rejected and the equation has no solution.

Can you cross-multiply to solve a rational equation?+

Only when the equation is exactly one fraction equal to one fraction, like 2/(x − 1) = 3/(x + 2). Cross-multiplying is a shortcut for multiplying both sides by both denominators. If there are extra terms, use the LCD method instead.

Why do you have to check answers to rational equations?+

Multiplying by an expression containing the variable can introduce answers that make the original denominators zero. Dividing by zero is undefined, so those answers can't be solutions. Checking catches them.

What is the difference between a rational expression and a rational equation?+

A rational expression is a fraction with polynomials, like (x + 1)/(x − 2), which you simplify. A rational equation sets rational expressions equal, like (x + 1)/(x − 2) = 3, which you solve for x.

Related lessons

Rational ExpressionsSimplifying Rational ExpressionsLinear Equations with FractionsRational Inequalities

Calculators for this topic

Rational Equations Calculator