Solving rational equations (incl. extraneous solutions)
Solving Rational Equations
Solving Rational Equations is a topic in Rational Functions in the California Common Core State Standards. It is aligned to Standard A-REI.2, which requires students to solve rational equations in one variable and give rise to and identify extraneous solutions.
Rational equations are solved by clearing denominators with the LCD, then rejecting extraneous solutions that make a denominator zero.
Theory
A rational equation has variables in a denominator. To solve:
- Find the least common denominator (LCD).
- Multiply every term by the LCD to clear fractions.
- Solve the resulting equation.
- Reject extraneous solutions that make a denominator zero.
Clear the denominators:
How to solve
- Factor denominators and find the LCD.
- Multiply every term by the LCD.
- Solve the resulting polynomial equation.
- Check each solution against the original denominators.
Cross-multiply.
| \(x+1\) | \(=\) | \(2(x-1)\) |
| \(x+1\) | \(=\) | \(2x-2\) |
| \(x\) | \(=\) | \(3\) |
Multiply by \((x-3)\).
| \(x\) | \(=\) | \(3+2(x-3)\) |
| \(x\) | \(=\) | \(2x-3\) |
| \(x\) | \(=\) | \(3\) |
But \(x=3\) makes a denominator zero — no solution.
Multiply every term by \(3x\).
| \(6+x\) | \(=\) | \(3x\) |
| \(6\) | \(=\) | \(2x\) |
| \(x\) | \(=\) | \(3\) |
Multiplying by a variable expression can introduce values that make a denominator zero — extraneous solutions that must be rejected.
Common pitfalls
Frequently asked questions
How do you solve a rational equation?
Multiply every term by the LCD to clear fractions, then solve.
What is an extraneous solution?
A value the algebra produces that makes a denominator zero and must be rejected.
Why do extraneous solutions appear?
Multiplying by a variable expression can introduce invalid values.
Do you always need to check?
Yes — check every solution in the original equation.