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) + 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)