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
- Solve rational equations and check excluded values
- Use solving rational equations in symbolic and graph-based problems
- Check common mistakes before finalizing an answer
Worked example
Problem. Solve 3/x = 1/2.
- Worked Example: First identify exactly what the question is asking: Solve 3/x = 1/2.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cross-multiply or multiply both sides by 2x.
- 3 * 2 = x.
- So x = 6.
- Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.
Answer: 6
Practice problems
1. Solve 3/x = 1/2.
Show solution
- Warm-up: First identify exactly what the question is asking: Solve 3/x = 1/2.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cross-multiply or multiply both sides by 2x.
- 3 * 2 = x.
- So x = 6.
- Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.
Answer: 6
2. Solve 4/x = 1/2.
Show solution
- Warm-up: First identify exactly what the question is asking: Solve 4/x = 1/2.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cross-multiply or multiply both sides by 2x.
- 4 * 2 = x.
- So x = 8.
- Check the result by substituting or estimating: the response should match 8 and make sense in the original problem.
Answer: 8
3. Solve 5/x = 1/2.
Show solution
- Core Practice: First identify exactly what the question is asking: Solve 5/x = 1/2.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Cross-multiply or multiply both sides by 2x.
- 5 * 2 = x.
- So x = 10.
- Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.
Answer: 10
4. After solving a rational equation, why check the original denominators?
Choices: to reject excluded values · to change every answer to positive · to find the graph color · to avoid using multiplication
Show solution
- Clearing denominators can introduce values that make the original denominator zero.
- Those values are not allowed.
- Checking prevents extraneous solutions.
Answer: to reject excluded values
5. After solving a rational equation, why check the original denominators? (variation 2)
Choices: to reject excluded values · to change every answer to positive · to find the graph color · to avoid using multiplication
Show solution
- Clearing denominators can introduce values that make the original denominator zero.
- Those values are not allowed.
- Checking prevents extraneous solutions.
Answer: to reject excluded values
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