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Inequality

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Source: — Problem Solving |

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by Mike@Magoosh » Wed Jan 22, 2014 12:57 pm
mariofelixpasku wrote:If s - 1/s < 1/t - t. Is s > t ?

A. s > 1
B. t > 0
Dear mariofelixpasku,
I'm happy to help. :-)

Here's a blog that is somewhat germane to this question:
https://magoosh.com/gmat/2013/gmat-quant ... qualities/

The curious thing to notice is that if s = 1/t, the two sides are equal. The fact that we can make them equal suggests we could make either one bigger if we liked.

The individual statements are not sufficient. I won't go through that, but when we specify only one variable, the other is entirely wide open.

Let's just focus on what happens when we combine the statements.
Statement #1: s > 1
Statement #2: t > 0
Well, following our inside above, if we pick s = 20 and t = 1/20, the two sides are equal. We could also make either side bigger. For example, pick s = 30 and t = 1/20. Then,
left = 30 - (1/30)
right = 20 - (1/20)
and left is larger. Now, pick pick s = 20 and t = 1/30. Then,
left = 20 - (1/20)
right = 30 - (1/30)
and right is bigger.
Even with both pieces of information in effect, we could make either side bigger or we could make the sides equal. We have absolutely no way to give a definitive answer to the prompt question. All the combined information is still insufficient. Answer = [spoiler](E)[/spoiler].

Does all this make sense?
Mike :-)
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by vipulgoyal » Wed Jan 22, 2014 9:10 pm
Hi Mike I believe OA should be C, please correct me if i am wrong

(s^2 - 1)/s < (1 -t^2)/t
by 1&2
since we got both s and t are positive we can cross multiply without changing the eneqality

t(s^2 -1) < s(1-t^2)
s^2t -t < s - st^2
s^2t + st^2 < s+t
st(s+t) < (s+t)
st < 1

since we know s>1 and t>0, t cant be more than s hence C
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by GMATGuruNY » Thu Jan 23, 2014 3:56 am
If s - 1/s < 1/t - t. Is s > t ?

A. s > 1
B. t > 0

Statement 1: s > 1
Since s>1, s - 1/s > 0.
To illustrate:
If s=2, then s - 1/s = 2 - 1/2 = 3/2.
If s=3/2, then s - 1/s = 3/2 - 2/3 = 5/6.
If s=10, then s - 1/s = 10 - 1/10 = 99/10.

Linking together 1/t - t > s - 1/s and s - 1/s > 0, we get:
1/t - t > s - 1/s > 0
1/t - t > 0.
For it to be true that 1/t - t > 0, it is not possible that t>1.
To illustrate:
If t = 2, then 1/t - t = 1/2 - 2 = -3/2. which is not positive.
If t = 3/2, then 1/t - t = 2/3 - 3/2 = -5/6, which is not positive.
If t = 10, then 1/t - t = 1/10 - 10 - -99/10, which is not positive.
Since s>1, and t CANNOT be greater than 1, s>t.
SUFFICIENT.

Statement 2: t > 0
Case 1: s=1 and t=1/2:
The constraint that s - 1/s < 1/t - t is satisfied:
1 - 1 < 2 - 1/2
0 < 3/2.
In this case, s>t.

Case 2: s=1/2 and t=1/2
The constraint that s - 1/s < 1/t - t is satisfied:
1/2 - 2 < 2 - 1/2
-3/2 < 3/2.
In this case, s=t.
INSUFFICIENT.

The correct answer is A.
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by Mike@Magoosh » Thu Jan 23, 2014 10:04 am
Hello all,
I must apologize. I misread the question. Not surprisingly, the brilliant Mitch is perfectly correct. The answer indeed should be [spoiler](A)[/spoiler].
Mike :-)
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by mariofelixpasku » Tue Feb 18, 2014 2:26 pm
is it possible to see such questions on the GMAT ?
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by [email protected] » Tue Feb 18, 2014 4:42 pm
Hi mariofelixpasku,

This DS question is rather "involved" (lots of little rules to keep track of, combined with inequalities and a YES/NO question), but all of the pieces that make up this question can (and likely will) show up on the GMAT. They might not all show up in same question though. Individually, none of the rules in this question is that tough to figure out, but seeing them all at once can seem "scary." The adaptive nature of the GMAT means that as you're consistently performing at a certain level, the test will adapt and give you questions at that level. If your goal is to score at a high level on Test Day, then you're going to have to get comfortable with Quant (and Verbal) questions that include layered concepts. When something tough-looking shows up, break it down into small pieces and work from there.

GMAT assassins aren't born, they're made,
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Last edited by [email protected] on Wed Feb 19, 2014 1:20 pm, edited 1 time in total.
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by GMATGuruNY » Wed Feb 19, 2014 6:41 am
mariofelixpasku wrote:is it possible to see such questions on the GMAT ?
This problem -- while tricky -- seems within the scope of the GMAT.
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by Abhishek009 » Wed Feb 19, 2014 9:07 am
mariofelixpasku wrote:If s - 1/s < 1/t - t. Is s > t ?

A. s > 1
B. t > 0
Plug in some values

Let S = 2 and t = -4

L.H.S.= s - 1/s

=> 2 - 1/2

=> 3/2

R.H.S. = 1/t - t

=>- (1/4) - (-4)

=> 4 - 1/4

=> 7/4



This satisfies the equation , hence we observe that S > T

Hence options A is our answer....
Abhishek
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by Matt@VeritasPrep » Wed Feb 19, 2014 5:38 pm
Another approach!

Before going to the two statements, manipulate the original inequality by adding to both sides:

1/s + 1/t > s + t

With a little algebra, this gives us

(s+t)/st > s + t

which is much easier to work with.

Now let's take statement 1, which tells us that s > 1. We can plug that into our first equation to get

(s+t)/st > (1+t)/t ≥ 1+t

Consider the second two parts of this inequality. With (1+t)/t ≥ 1+t, we consider three cases:

CASE 1: t is positive.

In this case, we divide both sides by (1+t), which is positive, to get 1/t ≥ 1, or 1 ≥ t. Since s > 1, this implies that s > t.

CASE 2: t is negative.

Since s > 1 and 0 > t, s > t.

SUFFICIENT!

Statement 2 is insufficient, but the explanations already given seem ample to me.
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