**Q: Condense: 3/2x + x/(2x-6)**

Answer:

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)** + 2x^{2} / **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 + 2x^{2}) / **2x(2x-6)**

I will rearrange the numeartor:

(2x^{2} + 6x – 18) / **2x(2x-6)**

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

2(x^{2} + 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:

(x^{2} + 3x – 9) / **x(2x-6)**

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

(x^{2} + 3x – 9) / (2x^{2} – 6x)