Simplify Rational Equations [Algebra II]

[hismajesty]

I am a bit confused on this problem, even though it appears to be faily simple. Any help is appreciated.

1-3x       2
------  + ----
x-6        2x+1

Here's what I've done so far..

I multiplied the fraction on the left of the addion sign by:

( 2x+1 )
(--------)
( 2x+1 )

And that on the right side by:

(x-6)
(----)
(x-6)

Which gives me:

(1-3x)(2x+1)       2(x-6)
----------------   + --------------
 (x-6)(2x+1)       (x-6)(2x+1)

After Foiling I've concluded that the answer is:

-x-6x^2-11
--------------
(x-6)(2x+1)

However, I'm sure this is incorrect as I started getting confused while looking at notes/doing what I thought was correct. Thanks in advance

[Yoni]

Almost correct.
You're correct up to (and including) this part:

Which gives me:

(1-3x)(2x+1)         2(x-6)
----------------   + --------------
 (x-6)(2x+1)         (x-6)(2x+1)

Now:
(1-3x)(2x+1) = 2x - 6x^2 + 1 - 3x = -6x^2 - x + 1
2(x-6) = 2x - 12
Adding them gives:
(1-3x)(2x+1) + 2(x-6) = -6x^2 - x + 1 + 2x - 12 = -6x^2 + x - 11

The final answer is:
-6x^2 + x - 11
------------------
 (x-6)(2x+1)
Which cannot be reduced further.

Basically you were correct except for the sign on the "x".

Edit: Formatting.

[hismajesty]

hmm, today in class I worked this problem with another student and we got a different answer. Here's what we/I did why is this not correct?

Again, the problem is:

1-3x      2
------ + ---- 
x-6       2x+1

As before I multiplied either side by the opposite sides denominator again giving me:

(1-3x)(2x+1)       2(x-6)                (1-3x)(2x+1) + 2(x-6)
----------------   + ---------------   =  ----------------------------
(x-6)(2x+1)         (2x+1)(x-6)                 (2x+1)(x-6)

I then crossed out the (2x+1) and the (x-6) leaving: -3x+3. Why is this incorrect?

[Yoni]

The "crossing out" part is incorrect.
You can't cross out if you have a sum, only if you have a product.

[hismajesty]

Yep, I noticed that and went back and corrected it, ended up getting the answer you gave. Thanks for the help.