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

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The discussion focuses on simplifying the expression (1-x)/(1+x) to arrive at (2/(x+1)) - 1. Participants suggest various methods, including rewriting 1-x as -(x+1) + 2 for simplification and using polynomial long division to achieve the same result. Another approach involves finding a common denominator by expressing 1 as (1+x)/(1+x) and simplifying the numerator. These techniques aim to clarify the steps needed to understand the equivalence presented in the textbook. Overall, the conversation emphasizes different strategies for simplifying rational expressions effectively.
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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.
 
Neat Trick. Thanks!
 
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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