If s - 1/s t ? (1) s 1 (2) t 0

Question

If  s - \frac{1}{s} < \frac{1}{t} - t , then is  s > t  ?

(1)  s > 1 
(2)  t > 0

Nidzo · Answer

We are told s - \frac{s}{1} < \frac{t}{1} - t . We can rearrange that to be: \frac{s^2-1}{s}<\frac{1-t^2}{t} (1) Because s>1 , we know then that t must be a positive fraction where 0t If t is negative, \frac{1-t^2}{t} will be negative because of the denominator. If t is greater than 1 , \frac{1-t^2}{t} will be negative because of the numerator. This leaves 01 , however, we do not know enough about s . As we do not know whether s is positive or negative we cannot cross multiply. However, we can plug in values to see if we can deduce that s>t or t~~t. INSUFFICIENT Answer A~~

Petunii · Answer

Hmmm I see a mistake here on cross multiplication that brings you nevertheless to the official correct answer. 
Can you please correct it and explain again how is Statement 1 Sufficient?

Petunii · Answer

Actually, I think you made a transcription error that made you compute a different problem.

First, it's written 1/t - t, where t could be negative ( (1 - t^2) / t where t < - 1 would yield a positive fraction).
Second, the rest of the explanation has the wrong elements from the beginning and the multiplication mistake that I previously mentioned.

Can you review and correct your explanation then?
And please correct me if I'm wrong!

Nidzo · Answer

Petunii  Well spotted. I wrote down the question incorrectly. It's been corrected.

Petunii · Answer

Thank you for your answer!

Unfortunately, I am still not convinced.
As I mentioned before, if t < -1 then the fraction would be positive, resulted from negative numerator divided by negative denominator.

Correct me if I'm wrong

MBAHOUSE · Answer

In my humble opinion, complicated question because you need to plug in numbers fast and work with some kind of projection. 
The question suggests C, but it’s an A.

Bunuel · Answer

Official Solution: If s - \frac{1}{s} < \frac{1}{t} - t , then is s > t ? First, let's rewrite the inequality as s + t < \frac{1}{s} + \frac{1}{t} . (1) s > 1 If 1 < s \leq t were true, then s + t would be the sum of two numbers greater than 1, hence more than 2. On the other hand, \frac{1}{s} + \frac{1}{t} would be the sum of two numbers less than 1, which would be a number less than 2. Therefore, if s < t were true, 1 < s \leq t would be greater than \frac{1}{s} + \frac{1}{t} , contradicting the given inequality. Consequently, 1 < s \leq t is not possible, and thus, s > t . Sufficient. (2) t > 0 Without any information on s , this statement is not sufficient. For example, consider s = -2 and t = 1 for a NO answer, and s = 1 and t = \frac{1}{2} for a YES answer. Answer: A

HoudaSR · Answer

We are given that s - 1/s < 1/t - t Which can also be written as, (s^2-1)/s < (1-t^2)/t Statement 1 tells us that s>1 Hence s^2>1 So, s^2-1>0 With this information, I know that (s^2-1)/s is positive and (1-t^2)/t is greater than something positive. Therefore, (1-t^2)/t is itself positive. (1-t^2)/t >0 translates to (1-t)(1+t)/t > 0 also to, (t-1)(t+1)/t < 0 Using the number line, we can tell that t<-1 or 01 as given in statement 1, s is definitely greater than t. Statement 1 is sufficient. Statement 2 tells us that t>0 but without information about s we can't tell if s>t. Statement 2 is not sufficient. Correct answer is A.

If s - 1/s < 1/t - t, then is s > t ? (1) s > 1 (2) t > 0

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