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Algebra 2 practice

How to Solve Rational Equations

Multiply every term by the least common denominator to clear the fractions, solve the equation that's left, then check each candidate in the original denominators — any answer that makes one zero is extraneous. Below: a worked example (including a no-solution case), the traps, and practice with full explanations.

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What you'll practice

Rational equations have variables in denominators. The method: multiply through by the least common denominator, solve what's left — and then CHECK each answer against the original denominators, because any "solution" that makes a denominator zero is extraneous and must be rejected. These questions drill the full loop, including the cases where every candidate dies.

Worked example

Solve xx−1=1x−1+2\frac{x}{x-1} = \frac{1}{x-1} + 2.

  1. Multiply every term by (x−1)(x-1): x=1+2(x−1)x = 1 + 2(x-1).
  2. Simplify: x=1+2x−2=2x−1x = 1 + 2x - 2 = 2x - 1.
  3. Solve: −x=−1-x = -1, so x=1x = 1.
  4. Check against the original: x=1x = 1 makes the denominator x−1=0x - 1 = 0 — undefined. The candidate is extraneous, so the equation has NO solution.

Common mistakes

  • Skipping the check: solving cleanly doesn't mean the answer survives — always test candidates in the original denominators.
  • Multiplying only some terms by the LCD: every term on both sides gets multiplied, including constants.
  • Rejecting a valid answer: a candidate is extraneous only if it zeroes a denominator — don't discard solutions just for being negative or fractional.

Try a few

1. Solve 5x=10x+3\frac{5}{x} = \frac{10}{x+3}.

  • x=3x = 3
  • x=−3x = -3
  • x=5x = 5
  • No solution

2. Solve 2x+13=1\frac{2}{x} + \frac{1}{3} = 1.

  • x=3x = 3
  • x=2x = 2
  • x=6x = 6
  • x=32x = \frac{3}{2}

3. Solve x+2x−2=3\frac{x+2}{x-2} = 3.

  • x=4x = 4
  • x=2x = 2
  • x=−4x = -4
  • No solution

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FAQ

How do you solve a rational equation?
Multiply every term by the least common denominator to clear the fractions, solve the resulting equation, then check each candidate against the original denominators and reject any that make one zero.
What is a rational equation?
An equation with a variable in at least one denominator, like x/(x−1) = 1/(x−1) + 2. The variable in the denominator is what creates the risk of extraneous solutions.
What is an extraneous solution?
An answer produced by the algebra that fails in the original equation — usually by making a denominator zero. Clearing denominators can create them, which is why the final check is part of the method, not optional.
Is Cramzoo free?
Cramzoo is free to start — every unit opens with free skills and no account is needed. This particular skill is part of Premium ($6/month or $30/year), which unlocks every skill plus the parent progress dashboard.
How is this different from IXL?
No punishing score. A wrong answer gives you a step-by-step explanation, not a number that drops — and every question is double-checked for accuracy before it reaches you.

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