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If a is not equal to b, is 1/(a-b) > ab ?

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If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 27 Feb 2013, 12:17
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If a is not equal to b, is 1/(a-b) > ab ?

(1) |a| > |b|
(2) a < b
[Reveal] Spoiler: OA
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 27 Feb 2013, 21:22
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mun23 wrote:
If a is not equal to b, is 1/(a-b) > ab ?

(1) |a| > |b|
(2) a < b


From F.S 1, let's assume a = -3 and b = -2. Thus, 1/(a-b) = -1 and a*b = 6. Thus, as -1<6, the answer to the question stem is No. Again, pick a = -3 and b = 2, and 1/(a-b) = -0.2, and a*b = -6. In this case we see that -0.2>-6, thus the answer to the question stem is a YES. Insufficient.

From F.S 2, lets again assume a = -3 and b = -2. Just as above we still get a NO. Again choosing the same set for a = -3 and b = 2, we get a YES to the question stem. Insufficient.

Combining both, we know that b-a>0 and mod(a)-mod(b)>0. Thus lets choose a=-7 and b=-2. We get 1/(a-b) = -0.2 and a*b = 14. Thus a NO. Again, choosing b=3 and a=-5, we get a YES . Insufficient.

Basically, the two fact statements given together mean that (a+b)<0. It's because from F.S 1, we get a^2-b^2>0 or (a-b)*(a+b)>0. We have from F.S 2 that a-b<0. Thus, (a+b) has to be negative.

E.
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 01 Jan 2014, 22:44
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Value substitution is good to solve this:
1. |a|>|b| we can say a can not be zero bcz mod of b will always be positive or equal to zero thus a must be anything but not zero.

We can do value substitution to test all cases:
| a | b | a-b | 1/(a-b) | ab | Pass/ Fail for option (1)
| -3 | -2 | -1 | -1 | 6 | Fail
| -3 | 2 .| -5 | -1/5 | -6 | Pass
| 3 .| 2 | 1 | 1 | 6 | Fail
| 3 .| -2 | 5 | 1/5 | -6 | Pass
| -3 .| 0 | -3 | -1/3 | 0 | Fail
| 3 .| 0 | 3 | 1/3 | 0 | Pass

Multiple Pass / Fail inconsistent result, option one not sufficient.

Option (2) a < b Not sufficient.

Combine option 1 + 2
a<b
| a | b | a-b | 1/(a-b) | ab | Pass/ Fail for option (1)
| -3 | -2 | -1 | -1 | 6 | Fail
| -3 | 2 | -5 | -1/5 | -6 | Pass
| -3 | 0 | -3 | -1/3 | 0 | Fail

Again inconsistent result, thus both option also not sufficient.

Answer E.
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 18 Feb 2014, 02:10
(1) |a| > |b| Clearly IS.
Look at this:

a > b
-a > b
- a > -b
a > -b

Would give you various answers for the YES/NO Question. IS!

(2) a<b. Here, a could be 1 and b 2. then we had 1/-1 = -1 and 1 * 2 = 2. Hence 1/(a-b) < a*b. But if a = -1 and b = 2 then 1/(a-b) = -1/3 and a*b = -1 * 2 = -2. Thus 1/(a+b) > a*b. IS.

Answer E.
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 26 Apr 2014, 07:26
So let's see. I think fastest way is to pic numbers. Statement 1, let's first say a=-2, b=1 then we have a YES answer. Let's also say that a=2 and b=1 then we have a NO answer. Insufficient. Statement 2, we can use a=2 and b=1 again for a YES answer. For a NO answer we could use b=3 and a=1. Insufficient. Both statements together we have that we can still use a=-2 and b=1 for a YES answer. Additionally, we could also have that both 'a' amd 'b' are negative. As in a=-3 and b=-2, giving a NO answer.

Therefore answer is E
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 30 Apr 2014, 12:07
Hey Karishma & Bunuel,
Is there a faster way to solve this problem? I tried picking numbers but it took me more than 2 mins to arrive at the answer.
Thanks,
-Prasoon
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 30 Apr 2014, 23:58
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prsnt11 wrote:
Hey Karishma & Bunuel,
Is there a faster way to solve this problem? I tried picking numbers but it took me more than 2 mins to arrive at the answer.
Thanks,
-Prasoon


For this problem I'd still advice to use number plugging at one point or another.

If a is not equal to b, is 1/(a-b) > ab ?

(1) |a| > |b|. This statement implies that a is further from 0 then b. We can have 4 cases:

--------0--b--a--
-----b--0-----a--
--a-----0--b-----
--a--b--0--------

For the second case the LHS is positive, while RHS is negative: 1/(a-b) > ab;
For the fourth case the LHS is negative, while RHS is positive: 1/(a-b) < ab.

Two different answers. Not sufficient.

(2) a < b --> a - b < 0. The LHS is negative:

If a=-2 and b=1, then (1/(a-b)=-1/3) > (ab=-2);
If a=-2 and b=-1, then (1/(a-b)=-1) < (ab=2).

Two different answers. Not sufficient.

(1)+(2) We can have only the third or fourth cases from (1):

--a-----0--b-----
--a--b--0--------

We can use the same example as for (2):
If a=-2 and b=1, then (1/(a-b)=-1/3) > (ab=-2);
If a=-2 and b=-1, then (1/(a-b)=-1) < (ab=2).

Two different answers. Not sufficient.

Answer: E.

Hope it helps.
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Re: If a is not equal to b, is 1/(a-b) > ab ? [#permalink] New post 22 May 2014, 01:49
Alternative approach

Is 1/(a-b) - ab>0?

Is (a-b)(1-ab) / (a-b)> 0?

Statement 1 we have that a^2 - b^2 >0, or (a+b)(a-b) > 0

But still insufficient, we don't know anything about 'ab'

Statement 2 we have that a-b<0

Same here, we are missing information regarding the sign of (1-ab)

Both together

Since (a+b)(a-b)>0 and a-b<0, then a+b>0

Therefore a<0, but we don't know about 'b' hence again impossible to figure out sign of (1-ab)

Answer: E
Re: If a is not equal to b, is 1/(a-b) > ab ?   [#permalink] 22 May 2014, 01:49
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