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Learn Index · Patterns and Algebra Grade 11–12

Rational equations

Equations with variables in denominators.

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Equations with fractions

A rational equation has variables in a denominator:

  • 3/x + 1/2 = 2
  • (x + 1)/(x − 3) = 5
  • 1/(x − 1) + 2/(x + 1) = 4/(x² − 1)

Strategy

  1. Factor denominators and state restrictions.
  2. Multiply every term by the lowest common denominator to clear fractions.
  3. Solve the resulting equation.
  4. Check each answer against the restrictions.

Example 1

3/x + 1/2 = 2, x ≠ 0

  • LCD = 2x
  • 2x(3/x) + 2x(1/2) = 2x(2)
  • 6 + x = 4x
  • x = 2 ✓ (allowed)

Example 2: cross-multiplying

(x + 1)/(x − 3) = 5, x ≠ 3

  • x + 1 = 5(x − 3)
  • x + 1 = 5x − 15
  • x = 4 ✓

Extraneous solutions

Multiplying by an expression with a variable can introduce answers that don’t work. An extraneous solution makes an original denominator zero.

Example: x/(x − 2) = 2/(x − 2), x ≠ 2

  • Multiply by (x − 2): x = 2
  • But x = 2 is restricted → no solution.

Example: 1/(x − 1) + 2/(x + 1) = 2/(x² − 1)

  • Restrictions: x ≠ ±1
  • LCD: (x − 1)(x + 1)
  • (x + 1) + 2(x − 1) = 2
  • 3x − 1 = 2 → x = 1
  • x = 1 is restricted → extraneous; no solution.

Equations that become quadratic

Example: x + 6/x = 5, x ≠ 0

  • × x: x² + 6 = 5x
  • x² − 5x + 6 = 0 → (x − 2)(x − 3) = 0
  • x = 2 or x = 3 (both allowed)

Work problems

If A can do a job in a hours and B in b hours, together:

1/a + 1/b = 1/t

Example: One pipe fills a pool in 6 h; another in 3 h. Together?

1/6 + 1/3 = 1/t → 1/2 = 1/t → t = 2 hours

Worked examples

Example 1: 4/x = 2/3 → 2x = 12 → x = 6.

Example 2: 5/(x + 1) = 2 → 5 = 2x + 2 → x = 1.5.

Example 3: Painter A takes 4 h, B takes 12 h. Together? 1/4 + 1/12 = 1/3 → 3 hours.

Common mistakes

  • Not multiplying every term by the LCD.
  • Skipping the restriction check.
  • Cross-multiplying when there are more than two fractions or extra terms.

Try it yourself

  1. 6/x = 3
  2. 2/(x − 1) = 4/(x + 1)
  3. x − 8/x = 2
  4. 3/(x − 3) = x/(x − 3)
  5. Ann mows a lawn in 2 h; Ben in 3 h. Together?

Answers: 1) 2 2) 3 3) 4 or −2 4) no solution (x = 3 is restricted) 5) 1.2 h (72 minutes)

Quick recap

  • Multiply by the lowest common denominator to clear fractions.
  • Check answers against restrictions to reject extraneous solutions.
  • Work problems use 1/a + 1/b = 1/t.