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May 22nd, 2015, 04:24 PM   #1
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How to simplify these two fractions?

$\displaystyle \frac{10}{x-3}+\frac{15}{3-x}$

I got the common denominator $\displaystyle 4x-9$ by multiplying the two denominators together. Then I multiplied the numerators like $\displaystyle 10(3-x)$ and $\displaystyle 15(3-x)$ so the equation ended up as so:

$\displaystyle \frac{30-10x+15x-45}{4x-9}$

and added and subtracted in the numerator to get

$\displaystyle \frac{5x-15}{4x-9}$

And now I'm just puzzled. I can see that three can divide into the numerator and that 9 in the denominator is divisible by 3, too, but that 4 there messes it all up. I might have made a calculation error somewhere, but I must have made the same error three times in a row if that's the case (I kept trying to see if I ended up with easier-to-work-with numbers, so I retried three times), so sorry for any carelessness. The answer is $\displaystyle \frac{-5}{x-3}$.
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May 22nd, 2015, 04:29 PM   #2
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Firstly, $(x - 3)(3 - x) \neq 4x - 9$.

Secondly, it might be easier to note that $3 - x = -(x - 3)$ and find a common denominator that way.
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May 22nd, 2015, 04:44 PM   #3
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Quote:
Originally Posted by Azzajazz View Post
Firstly, $(x - 3)(3 - x) \neq 4x - 9$.

Secondly, it might be easier to note that $3 - x = -(x - 3)$ and find a common denominator that way.
Oh jeez. I multiplied the x and -x to get -2x. I thought it looked strange. My bad. I'll try again, and thanks for pointing that out about -(x - 3).

Edit: Got $\displaystyle \frac{5}{3-x}$, so I got it. Thanks, Azzajazz! I'll be looking more carefully for denominators like this in the future.

Last edited by Rexan; May 22nd, 2015 at 04:47 PM.
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May 22nd, 2015, 06:22 PM   #4
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Quote:
Originally Posted by Rexan View Post
$\displaystyle \frac{10}{x-3}+\frac{15}{3-x}$
You can rewrite this way: 10 / (x-3) - 15 / (x-3)
so: -5 / (x-3)
And that can be rewritten: 5 / (3-x)
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May 22nd, 2015, 09:01 PM   #5
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Quote:
Originally Posted by Denis View Post
You can rewrite this way: 10 / (x-3) - 15 / (x-3)
so: -5 / (x-3)
And that can be rewritten: 5 / (3-x)
Wow, looks like you were able to snap that answer up pretty quick. Thanks for posting!
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