Basic Algebra/Rational Expressions and Equations/Adding and Subtracting When the Denominators are Different

Lesson

If you have to add two rational fractions with different denominators, as the first step, you have to find the LCM:

  3   +   3
 x+1     x-1

  LCM = (x+1)(x-1)

Now divide the LCM by both denominators and multiply by their respectives numerators:

  (x+1)(x-1) / (x+1) = (x-1) . (3) = 3x-3
  (x+1)(x-1) / (x-1) = (x+1) . (3) = 3x+3

The sum of the two results would be the new nominator:

  3x-3+3x+3 =
  (x+1)(x-1)

     6x
 (x+1)(x-1)

This is another example:

  6x   +   9x
 2x-6    x2-6x+9

We factorize both denominators and find the LCM

  2x-6 = 2(x-3)
  x2-6x+9 = (x-3)2
  LCM = 2(x-3)2

Now we divide and multiply:

  2(x-3)2 / 2(x-3) =
  2x2-12x+18 / 2x-6 = x-3
  (x-3) . 6x = 6x2-18x

  2(x-3)2 / (x-3)2 =
  2x2-12x+18 / x2-6x+9 = 2
  (2) . (9x) = 18x

We add the results to obtain the nominator; the denominator is the LCM:

  6x2-18x+18x = 
    2(x-3)2

   6x2
 2(x-3)2

We can factorize the nominator to simplify the result:

  2(3x2) =
  2(x-3)2

  3x2
 (x-3)2

Practice Problems

Easy

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Medium

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Hard