Rational Equation Solver
Solve rational equations (fractions with variables) step-by-step. This solver finds domain restrictions, shows the actual cleared polynomial equation, solves it, and removes extraneous solutions — with a labeled number-line visual, exact fraction answers when they exist, and a summary you can copy.
Background
A rational equation contains one or more rational expressions like 1/(x−2). You can multiply both sides by a common denominator to eliminate fractions — but you must exclude any value that makes a denominator 0. That’s where extraneous solutions come from.
How to use this solver
- Type your equation using x, fractions, and parentheses.
- Click Solve to see domain restrictions, the cleared polynomial equation, and solutions.
- Any value that makes a denominator 0 is excluded — the number line labels exactly which values and why.
- Use “extraneous check” to see which candidates fail in the original equation.
How this solver works
Domain restrictions come from each side’s own denominator — found before any algebra, since a value that makes either side undefined can never be a valid solution.
Clearing denominators multiplies both sides by the product of the denominators (an LCD-style factor), producing one polynomial equation set equal to 0 — shown as the “cleared equation.”
The real roots of that polynomial are the candidate solutions.
Each candidate is checked two ways: does it land on an excluded value, and does it actually satisfy the ORIGINAL equation? Either failure makes it extraneous.
Two special results can appear: a cleared equation of “0 = 0” means every value in the domain is a solution (infinitely many), and one like “1 = 0” means there’s no solution at all.
Formula & Equation Used
A rational equation often looks like: A(x)/B(x) = C(x)/D(x)
Domain restrictions: B(x) ≠ 0, D(x) ≠ 0
Clearing denominators (LCD idea): multiply both sides by a common denominator so the fractions disappear, leaving a single polynomial equation.
Exact fractions: when a decimal solution has a clean rational form (denominator ≤ 24), the calculator also shows it as a fraction — e.g. x ≈ 0.1429 alongside x = 1/7.
Example Problems & Step-by-Step Solutions
Example 1 — Two fractions (LCD)
Solve 1/(x-1) + 2/(x+1) = 1.
Steps: Restrictions x≠1, x≠−1. Multiply by (x−1)(x+1) to clear fractions, solve the resulting equation, then check each candidate in the original.
Example 2 — Proportion-style (fraction answer)
Solve (x+2)/(x-1) = (x-3)/(x+1).
Steps: Clearing denominators gives 7x = 1, so x = 1/7. That's a clean fraction, so the calculator shows both x ≈ 0.1429 and x = 1/7.
Example 3 — Extraneous solution appears
Solve x/(x-3) = 2 + 3/(x-3).
Steps: Restriction x≠3. Clearing denominators: x = 2(x−3)+3 = 2x−3, so x=3 — but that's excluded. The only candidate is extraneous, so there's no solution.
Example 4 — Identical sides (infinite solutions)
Solve 1/(x-2) = 1/(x-2).
Steps: Restriction x≠2. Clearing denominators leaves 0 = 0 — always true. Every real x except 2 is a solution.
Frequently Asked Questions
What is an extraneous solution?
A value that appears after clearing denominators but fails in the original equation — often because it makes a denominator 0 or changes the equation when multiplied by 0.
Why do we need domain restrictions?
Because rational expressions are undefined where the denominator equals 0 — those values must be excluded from the solution set no matter what the algebra produces.
Do I always need the "LCD"?
You need a common denominator that clears every fraction. The LCD is the smallest choice, but multiplying by a larger common denominator still works, as long as you track restrictions.
Does this solve complex solutions?
This version focuses on real solutions and real-domain restrictions. A cleared equation like x²+1=0 has no real roots, even though it has complex ones.
What does a cleared equation of "0 = 0" or "1 = 0" mean?
"0 = 0" means both sides were the same rational expression all along, so every value in the domain is a solution. "1 = 0" (or any nonzero number = 0) is a contradiction — no value of x can ever make it true.
Why does a solution sometimes show a fraction next to a decimal?
When a solution has a clean rational form with a small denominator (24 or less), the calculator shows the exact fraction — like 1/7 — alongside the rounded decimal, since the decimal alone can hide the exact value.
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