OG 12 Problem 208

[menacel]

I've searched for this problem and found a previous thread, but the answers didn't really help me out, see here: https://www.beatthegmat.com/if-x-not-equ ... tml#264953

The question is:

Given: (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

Stole this


[spoiler]A (x+1/x-1)^2[/spoiler]

I understand most of the simplification, but what I don't understand is when we introduce an "X" out of no where to eliminate the compound fractions. I must have missed this in grade school, but can anyone explain where it comes from?

[ ((1/x)+1) / ((1/x)-1)]^2 --> [ (x*((1/x)+1)) / (x*((1/x)-1) ] ^2 ==> ( 1+x / 1-x) ^2

[GMATGuruNY]

Much easier to plug in. Plug in x=2.

Then 1/x = 1/2.

When 1/2 is used in place of x in the given expression, we get:

[ (1/2 + 1) / (1/2 - 1) ]²

= [ (3/2)/(-1/2) ]²

= (-3)²

= 9.

Now we plug x=2 into all the answer choices to see which yields our target of 9.

Only answer choice A works:
[(x+1)/(x-1)]² = [(2+1)/(2-1)]² = 3² = 9.

A quick look at the other answer choices will show that none of them will yield 9 when we plug in x=2.

The correct answer is A.

[menacel]

Thanks, that's how I got by when I solved it.

But the OG's answer was a tad bit confusing for me.

[GMATGuruNY]

Thanks, that's how I got by when I solved it.

But the OG's answer was a tad bit confusing for me.

I discourage my students from reading the math explanations in the OG, which frequently do not offer the best approach.

[Anurag@Gurome]

I understand most of the simplification, but what I don't understand is when we introduce an "X" out of no where to eliminate the compound fractions. I must have missed this in grade school, but can anyone explain where it comes from?

[ ((1/x)+1) / ((1/x)-1)]^2 --> [ (x*((1/x)+1)) / (x*((1/x)-1) ] ^2 ==> ( 1+x / 1-x) ^2

That's because multiplication of the numerator and denominator of a fraction by the same nonzero quantity doesn't change the value of the fraction. Hence, multiplication with x doesn't change the value of the fraction, also it helps us to convert (1/x) to 1 in both the numerator and denominator.

[ahadi]

I discourage my students from reading the math explanations in the OG, which frequently do not offer the best approach.[/quote]

Hi,

Please advise which explanations for the PS in the OG do you recommend. I agree the OG often provides long-winded answers to most of the questions.

[smackmartine]

IMO A
(x+1/x -1)^2

Replacing x by 1/x we get
[(1+x)/(1-x)]^2 as (1-x)^2 = (x-1)^2 , we can rewrite the expression

(x+1/x -1)^2
So,A

[cans]

IMO A