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
- Clear denominators
- Solve rational equations
- Check restrictions
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.
- Multiply both sides by x - 1.
- 12 = 3(x - 1).
- 4 = x - 1, so x = 5.
Answer: 5
Practice problems
1. Solve x/5 = 9.
Show solution
- Warm-up: First identify exactly what the question is asking: Solve x/5 = 9.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Multiply by 5.
- Check the result by substituting or estimating: the response should match 45 and make sense in the original problem.
Answer: 45
2. Solve 18/(x + 2) = 6.
Show solution
- Core Practice: First identify exactly what the question is asking: Solve 18/(x + 2) = 6.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- 18 = 6(x + 2).
- Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.
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
- Challenge: First identify exactly what the question is asking: Why check rational equation solutions?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Excluded values are not allowed.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: They can make a denominator zero
4. Solve 10/x = 2.
Show solution
- Core Practice: First identify exactly what the question is asking: Solve 10/x = 2.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Multiply both sides by x, then divide by 2.
- Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.
Answer: 5
5. Solve (x + 1)/3 = 5.
Show solution
- Core Practice: First identify exactly what the question is asking: Solve (x + 1)/3 = 5.
- Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
- Multiply by 3, then subtract 1.
- Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.
Answer: 14
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