Algebraic Fractions
When adding or subtracting algebraic fractions, the first thing to do is to put them onto a common denominator (by cross multiplying).

e.g.      1      +      4    
       (x + 1)      (x + 6)

= 1(x + 6) + 4(x + 1)
       (x + 1)(x + 6)

= x + 6 + 4x + 4
   (x + 1)(x + 6)

=    5x + 10    
  (x + 1)(x + 6)

Solving equations
When solving equations containing algebraic fractions, first multiply both sides by a number/expression which removes the fractions.

Example:
Solve    10    - 2   =  1
         (x + 3)     x

multiply both sides by x(x + 3):
\ 10x(x + 3) - 2x(x + 3) = x(x + 3)
       (x + 3)            x

\ 10x - 2(x + 3) = x² + 3x      [after cancelling]
\ 10x - 2x - 6 = x² + 3x
\ x² - 5x + 6 = 0
\ (x - 3)(x - 2) = 0
\ either x = 3 or x = 2