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Rational Equations

A free College Algebra lesson from the “Rational Expressions and Equations” unit, with a worked example and practice problems including step-by-step solutions.

Rational equations can often be solved by multiplying both sides by the least common denominator. Solutions must be checked against original restrictions.

What you'll learn

Why it matters: Average speed, joint work, mixtures, and concentration problems often lead to rational equations. Clearing denominators makes the equation solvable, but restrictions still decide whether the candidate answer is valid.

Worked example

Problem. Solve 12/(x - 1) = 3.

  1. Multiply both sides by x - 1.
  2. 12 = 3(x - 1).
  3. 4 = x - 1, so x = 5.

Answer: 5

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. Solve x/5 = 9.

Show solution
  1. Multiply by 5.

Answer: 45

2. Solve 18/(x + 2) = 6.

Show solution
  1. 18 = 6(x + 2).

Answer: 1

3. Why check rational equation solutions?

Choices: They can make a denominator zero · They are always negative · They never graph · They always have one solution

Show solution
  1. Excluded values are not allowed.

Answer: They can make a denominator zero

4. Solve 10/x = 2.

Show solution
  1. Multiply both sides by x, then divide by 2.

Answer: 5

5. Solve (x + 1)/3 = 5.

Show solution
  1. Multiply by 3, then subtract 1.

Answer: 14

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