Algebraic fractions

Algebraic fractions follow the same rules as number fractions. Factorise first, cancel common factors, and use a common denominator to add or subtract.

Key facts

  • Factorise top and bottom before cancelling.
  • Only cancel factors, never single terms.
  • Add or subtract with a common denominator.
  • To solve an equation with fractions, multiply every term by the common denominator.

Worked examples

Example 1

Simplify (x² − 9)/(x² + 5x + 6).

  1. Top: (x + 3)(x − 3). Bottom: (x + 3)(x + 2).
  2. Cancel (x + 3). (x − 3)/(x + 2)

Answer: (x − 3)/(x + 2)

Example 2

Solve 3/(x + 1) + 2/x = 2.

  1. Multiply by x(x + 1). 3x + 2(x + 1) = 2x(x + 1)
  2. 5x + 2 = 2x² + 2x 2x² − 3x − 2 = 0
  3. (2x + 1)(x − 2) = 0 x = 2 or x = −½

Answer: x = 2 or x = −½