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
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