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Rational Equations

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

Rational equations contain fractions with variables. Multiply by the least common denominator to clear fractions, then solve and check that no denominator becomes zero.

What you'll learn

Why it matters: Average-speed problems, joint-work problems, and concentration mixes solve rational equations. Multiplying through clears the denominators — then always check that no answer makes the original denominator zero, since those are extraneous solutions.

Worked example

Problem. Solve x/3 = 5.

  1. Multiply both sides by 3.
  2. x = 15.
  3. Check: 15/3 = 5.

Answer: 15

Practice problems

1. Solve x/4 = 7.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve x/4 = 7.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Multiply by 4.
  4. Check the result by substituting or estimating: the response should match 28 and make sense in the original problem.

Answer: 28

2. Solve 3/x = 1 when x is not 0.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Solve 3/x = 1 when x is not 0.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. 3/x = 1 means x = 3.
  4. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Solve x/5 + 2 = 8.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve x/5 + 2 = 8.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Subtract 2.
  4. x/5 = 6.
  5. Check the result by substituting or estimating: the response should match 30 and make sense in the original problem.

Answer: 30

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Solve 12/(x - 1) = 3.
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Multiply by x - 1.
  4. 12 = 3(x - 1).
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. Why must rational equation solutions be checked?

Choices: They can make a denominator zero · They always have two answers · They cannot be graphed · They are never real

Show solution
  1. Challenge: First identify exactly what the question is asking: Why must rational equation solutions be checked?
  2. Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
  3. Excluded values are not allowed.
  4. 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

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