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If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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27 Dec 2012, 05:45

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\((\frac{x+1}{x-1})^2\)

If x#0 and x#1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to

A. \((\frac{x+1}{x-1})^2\)

B. \((\frac{x-1}{x+1})^2\)

C. \(\frac{x^2+1}{1-x^2}\)

D. \(\frac{x^2-1}{x^2+1}\)

E. \(-(\frac{x-1}{x+1})^2\)

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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27 Dec 2012, 05:56
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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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Bunuel wrote:

[b]\((\frac{x+1}{x-1})^2\) \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\) .

How did 1-X in the denominator become X-1 in last step?

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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nikhil007 wrote:

Bunuel wrote:

[b]\((\frac{x+1}{x-1})^2\) \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\) .

How did 1-X in the denominator become X-1 in last step?

It's not (1-x) that became (x-1).

\((1-x)^2\) is simply rewritten as \((x-1)^2\).

Both \((1-x)^2\) and \((x-1)^2\) are essentially the same

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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nikhil007 wrote:

Bunuel wrote:

[b]\((\frac{x+1}{x-1})^2\) \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\) .

How did 1-X in the denominator become X-1 in last step?

\((1-x)^2 = (x-1)^2\). For example, \((1-4)^2 = (4-1)^2 = 9\).

The negative sign inside the bracket gets taken care of because of the square.

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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30 Jun 2013, 17:47

Does anyone have any similar questions this this one?

I like this problem.

Thanks,

Hunter

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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25 Jul 2013, 01:48

Bunuel wrote:

\((\frac{x+1}{x-1})^2\) If x#0 and x#1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to A. \((\frac{x+1}{x-1})^2\) B. \((\frac{x-1}{x+1})^2\) C. \(\frac{x^2+1}{1-x^2}\) D. \(\frac{x^2-1}{x^2+1}\) E. \(-(\frac{x-1}{x+1})^2\) \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\). Answer: A.

I don´t get \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2\).. could you please explain your steps in a few words?

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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25 Jul 2013, 02:12
sv3n wrote:

Bunuel wrote:

\((\frac{x+1}{x-1})^2\) If x#0 and x#1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to A. \((\frac{x+1}{x-1})^2\) B. \((\frac{x-1}{x+1})^2\) C. \(\frac{x^2+1}{1-x^2}\) D. \(\frac{x^2-1}{x^2+1}\) E. \(-(\frac{x-1}{x+1})^2\) \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\). Answer: A.

I don´t get \((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2\).. could you please explain your steps in a few words?

Step by step:

\((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2=(\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2\).

\((\frac{\frac{1+x}{x}}{\frac{1-x}{x}})^2=(\frac{1+x}{x}*\frac{x}{1-x})^2\)

\((\frac{1+x}{x}*\frac{x}{1-x})^2=(\frac{1+x}{1-x})^2\)

\((\frac{1+x}{1-x})^2=(\frac{x+1}{x-1})^2\)

Can you please tell me which step didn't you understand?

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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25 Jul 2013, 04:10

I do not understand the first of these four steps. -.- I understand that 1/x+1 is the same as 1+x/x, but why have you done it? I only get it if I see the result and go backwards..

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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25 Jul 2013, 06:10

Tried it several times again. I think I got it know.. thanks.

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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Walkabout wrote:

\((\frac{x+1}{x-1})^2\) If x#0 and x#1, and if x is replaced by 1/x everywhere in the expression above, then the resulting expression is equivalent to A. \((\frac{x+1}{x-1})^2\) B. \((\frac{x-1}{x+1})^2\) C. \(\frac{x^2+1}{1-x^2}\) D. \(\frac{x^2-1}{x^2+1}\) E. \(-(\frac{x-1}{x+1})^2\)

We can use a substitution method and process of elimination method to avoid the cumbersome calculations.

Say x = 2

x is replaced by 1/x. So, [(1/x + 1)/(1/x -1)] ^2 = [1+x/1-x] ^2

By substituting 2 we get, [(1+2)/(1-2)] ^2 = 9

Now, substitute x = 2 in the answer choices and an answer choice that gives the final answer as 9 is the correct answer.

Option A gives us 9.

Hence, A is the correct answer.

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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nikhil007 wrote:

How did 1-X in the denominator become X-1 in last step?

Rewrite the equation so that \(x\) appears as the first term in the equation:

\((1-x)^2 = (-x+1)^2\)

let's now rewrite the equation so that \(x\) is positive:

\((1-x)^2 = (-x+1)^2 = [ (-1) \cdot (x-1)]^2\)

the laws of exponents establish that \((a \cdot b)^n = a^n \cdot b^n\) which means that:

\((1-x)^2 = (-x+1)^2 = [ (-1) \cdot (x-1)]^2 = (-1)^2 \cdot (x-1)^2\)

notice that \((-1)^2 = -1 \cdot -1 = 1\) therefore:

\((1-x)^2 = (-x+1)^2 = [ (-1) \cdot (x-1)]^2 = (-1)^2 \cdot (x-1)^2 = 1 \cdot (x-1)^2 = (x-1)^2\)

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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20 Aug 2014, 13:50

Can't we just multiply the numerator and denominator by x, after substituing in (1/x)?

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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JackSparr0w wrote:

Can't we just multiply the numerator and denominator by x, after substituing in (1/x)?

We require to compute\(\frac{1}{x} + 1\) & \(\frac{1}{x} - 1\) before that

Refer Bunuel's method; done the very best

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If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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09 Dec 2014, 03:15

\((\frac{\frac{1}{x}+1}{\frac{1}{x}-1})^2\). I just expanded the formula out like this: (1/x+y)(1/x+y)/(1/x+y)(1/x-y) =(x+y)/(x-y) = which is the same as answer choice A when squared (?) Understand the other way mentioned above too but want to check if this is an alternative or if it's incorrect.

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You can also plug in numbers, such as x=2. Since x is being replaced by (1/x) we will replace x with (1/2). The original equation gives us a solution of 9. plugging in (1/2) into all of the answer solutions present us with A as the only correct answer.

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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20 Jan 2015, 08:03

This is what I did, but I am not sure if it is correct: [x+1]^2 / [x-1]^2 [(1/x)+1]^2 / [(1/x)-1]^2 [(1+x)/x]^2 / [(1-x)/x]^2 [(1+x)x]^2 / [(1-x)x]^2 (1+x)^2 / (1-x)^2 ANS A It would be easier to read alligned vertically. How do we do that?

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Re: If x#0 and x#1, and if x is replaced by 1/x everywhere in th [#permalink ]

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13 Feb 2016, 08:26

I replaced x with a value say 2 and then solved the whole problem.Got the answer correct. and its easy too without any confusion.

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