Simplifying fractional equations

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CalebsMomma

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Jul 13, 2009
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Ok, so now that you have all helped me figure out how to do the whole common denominator thing with fractions...
I have another question:
Here is the problem: Divide and express the result in its simplest form (the symbol between the two rational expressions is dividing):
2x/(x+5) / (x+5)/(x+1)=
2x/(x+5) * (x+1)/(x+5)=
2x(x+1)/(x+5)(x+5)=
(2x^2)+2x/(x^2)+5x+5x+25=
(2x^2)+2x/x^2+10x+25

So my question is, can I can parts of this equation, such as x^2 and it would become:
2+2x/10x+25
Which 2x can cancel and it would become:
2+1/(5x+25)= 3/(5x+25)
 
CalebsMomma said:
Ok, so now that you have all helped me figure out how to do the whole common denominator thing with fractions...
I have another question:
Here is the problem: Divide and express the result in its simplest form (the symbol between the two rational expressions is dividing):
2x/(x+5) / (x+5)/(x+1)=
2x/(x+5) * (x+1)/(x+5)=
2x(x+1)/(x+5)(x+5)=
(2x^2)+2x/(x^2)+5x+5x+25=
(2x^2)+2x/x^2+10x+25

So my question is, can I can parts of this equation, such as x^2 and it would become:
2+2x/10x+25
Which 2x can cancel and it would become:
2+1/(5x+25)= 3/(5x+25)
Click to expand...
1) Please fix your notation. It is very confusing. x + 2 / x - 5 is NOT the same as (x+2)/(x-5). Be very sure to write what you mean.

Examples:

3 + 5 / 5 - 1 = 3 + (5/5) - 1 = 3 + 1 - 1 = 3
(3 + 5) / (5 - 1) = 8 / 4 = 2

Think about this long and hard. Make sure it makes sense in your head.

2) "can parts" - Never EVER use this phrase or concept again.
3) "cancel" - Never EVER use this phrase or concept again.
4) An "equation" requires one of these: "=". You have an "expression".
5) This all may seem a bit like nit-picking, but if you learn to think and to write clearly, it WILL save you.

You may separate FACCTORS of 1 (one). That is ALL you may do. If you do not KNOW a factor is 1 (one), then don't do it. If it isn't a FACTOR, don't do it.

2x(x+1) has three factors. Can you name them?
2x^2 + 2x has the same three factors. 2x^2 is NOT a factor. 2x is NOT a factor.

\(\displaystyle \frac{\frac{2x}{x+5}}{\frac{x+5}{x+1}}\;\rightarrow\;\frac{2x}{x+5} \cdot \frac{x+1}{x+5}\;\rightarrow\;\frac{2x(x+1)}{(x+5)^{2}}\)

There are no common factors in numerator AND denominator. That's it. Stop kicking it around.
 
I put those things is parethesis because it confuses some people, I want to make sure that people know what I mean.
And what I meant is can things cancel and it is an equation it's equal to zero, I just never add it until I get everything worked down and then I add the =0.
So I just need to know can I make it simpler by canceling things out?
And I don't think that you are nit picking, I am just trying to figure this stuff out.
 
In division only things you can "cancel" out are the common factors.

If you do not have common factors - you cannot cancel out anything in division. Numerical example:

\(\displaystyle \frac{6\cdot 5}{5 \cdot 7}\)

5 is common factors above - so it can be cancelled out - so -

\(\displaystyle \frac{6\cdot 5}{5 \cdot 7} \, = \, \frac{6}{7}\)

Now if we had "addition" instead of "multiplication" - then we would not have common factors and no cancelling allowed:

\(\displaystyle \frac{6 \, + \, 5}{5 \cdot 7} \, = \, \frac{11}{35}\)

same rule applies for all through mathematics.
 
CalebsMomma said:
I put those things is parethesis because it confuses some people, I want to make sure that people know what I mean.
And what I meant is can things cancel and it is an equation it's equal to zero, I just never add it until I get everything worked down and then I add the =0.
So I just need to know can I make it simpler by canceling things out?
And I don't think that you are nit picking, I am just trying to figure this stuff out.
Click to expand...

There is no objection to additional parentheses for clarity. The problem is leaving out necessary parentheses.

You said "cancel" again. You have not gotten it out of your head. Trust me on this. Never, EVER "cancel" anything again, except maybe your expensive internet service provider. In mathematics, we have equivalence.

To use the extant example: \(\displaystyle \frac{6 \cdot 5}{5 \cdot 7}\)

This is an equivalent expression: \(\displaystyle \frac{6 \cdot 5}{7 \cdot 5}\) We have used what property of multiplication in the denominator?

This is an equivalent expression: \(\displaystyle \frac{6}{7} \cdot \frac{5}{5}\) We have used what properties of multiplication or division to accomplish this?

This is an equivalent expression: \(\displaystyle \frac{6}{7} \cdot 1\) We have used what property of multiplication or division to accomplish this?

This is an equivalent expression: \(\displaystyle \frac{6}{7}\) We have used what convention of mathematical notation to accomplish this?

There is NO "cancelling" going on in there. Never do it and never think it. You will unconfuse yourself if you forget about the word "cancel".
 
I totally agree - the words "cancel" & "cross-multiply" (which Deniss calls "criss-cross") should be out-of-bounds for math vocab.
 
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