Am I just wrong or are there two solutions to this problem?

Of course I am wrong, but I would like to know why- any help would be much appreciated.
5 2
----- _ ------
x-2 x+2


Now what I wanted to do and what I did do was:

5((x -2)(x+2)) _ 2((x -2)(x+2))
--------------- -----------------
(x-2) (x+2)

yielding: 5x + 10 -2x + 4= 3x + 14

This problem was on a test in the book I am using with and it had multiple choice answers. Only one answer had 3x + 14, so I chose it... but I was just lucky to get it right because the testmaker or psychometrician slacked off-s/he could have easily tricked me if 3x+14 alone were offered as a choice (or maybe by chance there are two answers and so that is why it was not offered).

The answer offered was 3x + 14
----------
x^2-4
The solution showed that the correct procedure was to cross multiply the denominators and numerators and then both denominators --that is nice and I am sure it is the only way but why is my approach wrong--is it wrong just because it's wrong or what? In other similar problems my approach would work wouldn't it? When is cross multiplying the best approach?
Thanks for any help guys.

Sorry it appears that the equation got misaligned--I will try again

There should be a 5 over the denominator of x-2 and that whole fraction is followed by a minus sign and then the fraction 2 over x +2

(5) ( 2)
___ (minus) ____
(x-2) (x+2)

5/[x-2] - 2/[x+2]
place over common denominator [x-2][x+2]
5[x+2]/{ [ x-2][x+2] } - 2[x-2] / { [ x-2][x+2] }
combine terms
{5[x+2]-2[x-2} / { [x-2][x+2] }
clear brackets
{5x+10-2x+4} / [x^2-4]
[3x+14] / [x^2-4] answer

Arthur

Thanks Arthur. I understood the procedure after I saw it in the book but I would really like to know why:
a)my approach was wrong?-so I don't screw up again
b) how do I now when cross multiplying is best?
c) when is a good time to use the approach I used?

Thanks. Sorry to be a pain in the butt with these silly questions.

Is my screw up connected to the fact that this problem had two terms connected by a minus sign? If so, maybe that is when I need to cross multiply.
Sorry guys I am just thinking aloud here--but with the 2x/3=3x-4, my approach of multiplying the numerators by (3)(4) works out--so maybe if there is an = sign I can do what I wanted to do but if there is a plus or minus I must cross multiply --is that right?

plugginaway said:

a)my approach was wrong?-so I don't screw up again
b) how do I now when cross multiplying is best?
c) when is a good time to use the approach I used?

Click to expand...

a) you ended up with: yielding: 5x + 10 - 2x + 4= 3x + 14
But you started off with an expression only; how did you ever come up with an equal sign?

b) something wrong here; cross multiplying is only possible if you have an equation

c) NEVER :wink:

Thanks Dennis. The problem is that I foolishly just started perfunctorily doing the problems--Most of the problems were like the second example I posted(I mistyped it too)where the problem was
2x/3 = 3x-1/4
with these types I could get the product of the denominators and multiply each of the numerators by that product---but with the type of eqaution that has a minus sign or plus sign for that matter, I have discovered I cannot do that.
I understand your point about the equal sign but with a regular fraction problem that has no "x" we have an expression without an equal sign right? And we would use an equal sign wouldn't we when answering it?
2/3 + 2/3 (Or maybe this isn't an expression--it is just a fraction) Boy, it is good to think these things through-sorry my stupidity is exposed
= 4/3
Tahnks for your help Denis

Maybe I miss understood the import of what you were referring to regarding the equal sign --I used an equal sign because I was combining like terms--wouldn't an equal sign be used in such a case?

plugginaway said:

Now what I wanted to do and what I did do was:

5((x -2)(x+2)) _ 2((x -2)(x+2))
--------------- -----------------
(x-2) (x+2)

yielding: 5x + 10 -2x + 4 = 3x + 14

Click to expand...

Ahhh...I see; I took that as if you were saying that was your final answer...my bad...

All I was trying to say is that the ANSWER to a simplification such as this problem
should not contain an equal sign: and the answer is (3x + 14) / (x^2 - 4);
you can use as many equal signs as you want to ARRIVE at the answer, like:
something
= something
= something
= (3x + 14) / (x^2 - 4)

you did not cross multiply! Your second line is wrong!

5/[x-2] - 2 /[x+2] cross multiply? NO!
Why not? WE ARE NOT CROSSING A EQUAL {=} SIGN!

You cross multiply a = sign , you must place over a common denominator!

Arthur

This is why the term cross-multiply has been taken out of the vocabulary - they create "lazy" confusion.

Anybody trying to propose cross-multiplying as a method - in this board - should go wash their fingers with soap.

Well Subhotosh, I'll wash me fingers often then :wink:

4 / (x+1) = 10 / (x+4) ; CROSSMULTIPLY(!):
4x + 16 = 10x + 10
x = 1

4 / (x+1) = 10 / (x+4)
4 / (x+1) - 10 / (x+4) = 0
Apply LCM....
Takes at least 10 times as long!

Subhotosh Khan said:

4 / (x+1) = 10 / (x+4) ; CROSSMULTIPLY(!): <<< Why not multiply both sides with (x+1)*(x+4)

Click to expand...

Why multiply both sides with (x+1)*(x+4) (when you can CROSSMULTIPLY!) ?

4 / (x+1) = 10 / (x+4) ; CROSSMULTIPLY(!): <<< Why not multiply both sides with (x+1)*(x+4)

Click to expand...

The point in question here is simply one of semantics. The math is identical; we really are just multiplying both sides of the equation by (x+1)(x+4), no matter how we say it. Unfortunately, this fact sometimes eludes students – thus leading to the sort of confusion that is giving our friend Plugginaway some difficulties.

To summarize:

“Cross multiplying” is an informal math expression (and process) that hides the real math steps involved. When we cross multiply (only across an equal sign!), what we are really doing is multiplying BOTH sides of an equation by BOTH denominators.

Plugginaway said:

Boy, it is good to think these things through-sorry my stupidity is exposed.

Click to expand...

Plugginaway, please don’t ever say or think that (“stupidity”)! The only thing you have “exposed” here is a desire to learn and the intelligence to ask questions. Thank you for your participation.