Condensing Fraction (involving variables)

Q:  Condense:  3/2x + x/(2x-6)


In order to add fractions, we need to have common denominators.  Remember, in order to get common denominators, we need to multiply by something.. Just because one denominator is 2x – 6 and the other is 2x, we cannot just subtract 6 to make them match.

So, the fraction on the left needs a (2x-6) and the fraction on the right needs a (2x)…  So, you have to multiply the fraction on the left by (2x-6) and the fraction on the right needs to be multiplied by (2x).  Remember:  multiply to the top and the bottom like so:

3/2x + x/(2x-6)

3(2x-6)/2x(2x-6) + x(2x)/(2x-6)(2x)

Now, simplify each numerator, leave the denominators alone:

6x – 18 / 2x(2x-6) + 2x2 / 2x(2x-6)

Now, since the denominators are the same, we can just combine the numerators to make one fraction.  The denominator goes unchanged:

(6x – 18 + 2x2) / 2x(2x-6)

I will rearrange the numeartor:

(2x2 + 6x – 18) / 2x(2x-6)

Now, let’s factor the top to see what we can cancel out (if we can):

2(x2 + 3x – 9) / 2x(2x-6)

The numerator cannot be factored any more, so we can cancel a 2 out of the top and out of the bottom to get:

(x2 + 3x – 9) / x(2x-6)

This is either the final answer, or you can multiply out the denominator to get:

(x2 + 3x – 9) / (2x2 – 6x)