General Simplification of (1+x)/(1-x)

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I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??
 

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You can write 1-x as -x-1+2=-(x+1)+2 and simplify.
That is a usual trick for fractions like yours.
 
remorris said:
I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??

mfb said:
You can write 1-x as -x-1+2=-(x+1)+2 and simplify.
That is a usual trick for fractions like yours.

An alternate approach is to divide - x + 1 by x + 1 using polynomial long division. If you don't know this technique, you can search Wikipedia using the search string "polynomial long division". Doing this, you get -1 + 2/(x + 1).
 
remorris said:
I know that (1-x)/(1+x) is equivalent to (2/(x+1)) - 1 via 'hand waving' in my textbook but i cannot figure out the steps to arrive at this result. Suggestions??

Yet another equivalent way to see this is to replace 1 by (1+x)/(1+x) and you now have a common denominator of 1 + x. Gathering terms in the numerator yields 1 - x.
 

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