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Solving Rational Equations

A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.

Clearing denominators is useful only if the final answers are checked against the original restrictions. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Rational functions model rates, constraints, efficiency, and quantities that change sharply near restricted inputs.

Worked example

Problem. Solve 9/x = 3/4. (x is an integer)

  1. Worked Example: First identify exactly what the question is asking: Solve 9/x = 3/4. (x is an integer)
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Cross-multiply: 9 * 4 = 3 * x.
  4. 36 = 3x, so x = 36/3 = 12.
  5. Check: 9/12 = 3/4. Valid (x is not 0).
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

Practice problems

1. Solve 9/x = 3/4. (x is an integer)

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 9/x = 3/4. (x is an integer)
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Cross-multiply: 9 * 4 = 3 * x.
  4. 36 = 3x, so x = 36/3 = 12.
  5. Check: 9/12 = 3/4. Valid (x is not 0).
  6. Check the result by substituting or estimating: the response should match 12 and make sense in the original problem.

Answer: 12

2. Solve 20/(x - 3) = 5. (x is an integer)

Show solution
  1. Multiply both sides by (x - 3): 20 = 5(x - 3).
  2. Divide by 5: 4 = x - 3, so x = 7.
  3. Check: x - 3 = 4 (not 0), and 20/4 = 5. Valid.

Answer: 7

3. Solve x/(x - 5) = 3/2. (x is an integer)

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x/(x - 5) = 3/2. (x is an integer)
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Cross-multiply: 2x = 3(x - 5).
  4. 2x = 3x - 15, so 15 = x.
  5. x = 15; denominator x - 5 = 10 (not 0). Valid.
  6. Check the result by substituting or estimating: the response should match 15 and make sense in the original problem.

Answer: 15

4. Solve 2/x + 3/x = 1. (x is an integer)

Show solution
  1. Combine the left side over the common denominator x: (2 + 3)/x = 5/x.
  2. So 5/x = 1, which gives x = 5.
  3. Check: x = 5 is not 0, and 2/5 + 3/5 = 5/5 = 1. Valid.

Answer: 5

5. Solve 3/(x - 2) = 2/(x - 6). (x is an integer)

Show solution
  1. Cross-multiply: 3(x - 6) = 2(x - 2).
  2. 3x - 18 = 2x - 4, so 1x = 14.
  3. x = 14; neither x - 2 = 12 nor x - 6 = 8 is 0. Valid.

Answer: 14

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