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Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|
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Author:  Bunuel [ 28 Feb 2014, 03:09 ]
Post subject:  Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

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

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

Author:  Bunuel [ 28 Feb 2014, 03:09 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be YES if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be NO. Not sufficient.

Answer: A.

Author:  anairamitch1804 [ 26 Jan 2017, 18:27 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Statement 1: a < b
Thus, a-b<0, implying that b-a>0.
Is 1/(negative) < positive?
YES.
SUFFICIENT.

Statement 2: 1 < |a - b|
It's possible that a-b = 2, implying that b-a = -2.
Plugging a-b=2 and b-a=-2 into 1/(a-b) < b-a, we get:
1/2 < -2?
NO.

It's possible that a-b = -2, implying that b-a = 2.
Plugging a-b=-2 and b-a=2 into 1/(a-b) < b-a, we get:
1/-2 < 2?
YES.

Since in the first case the answer is NO but in the second case the answer is YES, INSUFFICIENT.

The correct answer is A.

Author:  HKHR [ 01 Mar 2014, 02:21 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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



Statement 1: a < b
Thus, a-b<0, implying that b-a>0.
Therefore, LHS is negative and RHS is positive. Which implies LHS<RHS. Therefore Sufficient.

Statement 2: 1 < |a - b|
This statement implies 1<(a-b)<-1

Substitute a-b = 2 and a-b= -2 we get different answers for both cases. INSUFFICIENT.


Therefore answer is A.

Author:  lool [ 09 Mar 2014, 13:49 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Bunuel wrote:
SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be NO if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be YES. Not sufficient.

Answer: A.



for the second statement;

1 < |a - b| --> 1< a-b or a-b < -1

AND

a-b < -1 --> 1< b-a therefore LHS will be negative and RHS will be positive -->negative<positive. Sufficient.


what is wrong with my approach ?

Author:  Bunuel [ 09 Mar 2014, 14:02 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

lool wrote:
Bunuel wrote:
SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be NO if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be YES. Not sufficient.

Answer: A.



for the second statement;

1 < |a - b| --> 1< a-b or a-b < -1

AND

a-b < -1 --> 1< b-a therefore LHS will be negative and RHS will be positive -->negative<positive. Sufficient.


what is wrong with my approach ?


From \(1 < |a - b|\) we can have two cases:

A. \(1 < a-b\) --> \((\frac{1}{a - b}=positive) > (b - a=negative)\) --> answer YES.

B. \(a-b<-1\) --> \((\frac{1}{a - b}=negative) < (b - a=positive)\) --> answer NO.

Two different answers, hence insufficient.

Hope it's clear.

Author:  coolredwine [ 11 Mar 2014, 01:24 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

I did it like this:

We can reduce the given statement as:

1/(a-b)>(b-a) -> Multiplying both sides by (a-b), we get: 1>(a-b)(b-a)

Taking negative out from RHS: 1<(a-b)(a-b), which is what we have to prove.

Now, Statement 1: a<b -> a-b<0. Squaring both sides: (a-b)(a-b)<0. Hence, the answer would be No. Thus, sufficient.

Statement 2: 1<|a-b| -> 1<(a-b) or 1>(a-b). Thus insufficient.

Hence, answer is A.

Please do let me know if this is a good way to proceed with such questions.

Thanks.

Author:  Bunuel [ 11 Mar 2014, 01:46 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

coolredwine wrote:
I did it like this:

We can reduce the given statement as:

1/(a-b)>(b-a) -> Multiplying both sides by (a-b), we get: 1>(a-b)(b-a)

Taking negative out from RHS: 1<(a-b)(a-b), which is what we have to prove.

Now, Statement 1: a<b -> a-b<0. Squaring both sides: (a-b)(a-b)<0. Hence, the answer would be No. Thus, sufficient.

Statement 2: 1<|a-b| -> 1<(a-b) or 1>(a-b). Thus insufficient.

Hence, answer is A.

Please do let me know if this is a good way to proceed with such questions.

Thanks.


Unfortunately, most of it is wrong.

We cannot multiply 1/(a-b)>(b-a) by a-b, because we don't know its sign.
If a-b is positive, then we would have 1 > (b-a)(a-b);
If a-b is negative, then we would have 1 < (b-a)(a-b): flip the sign when multiplying by negative value.

Never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know its sign.

Next, you cannot square a-b<0 and write (a-b)^2<0. This is obviously wrong: the square of a number cannot be less than zero.

We can only raise both parts of an inequality to an even power if we know that both parts of the inequality are non-negative (the same for taking an even root of both sides of an inequality).

Adding/subtracting/multiplying/dividing inequalities: help-with-add-subtract-mult-divid-multiple-inequalities-155290.html

Hope this helps.

Author:  Lucky2783 [ 10 Apr 2015, 03:30 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Bunuel wrote:
The Official Guide For GMAT® Quantitative Review, 2ND Edition

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

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

Data Sufficiency
Question: 120
Category: Arithmetic; Algebra Arithmetic operations; Inequalities
Page: 161
Difficulty: 650


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1/(a - b) > b - a ?

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

1/(a - b) > b - a ?

\(\frac{1+(a-b)^2}{(a-b)}\) > 0 ? we can see that numerator is always +ive . all we need to know is a>b?

(1) a < b ; then \(\frac{1+(a-b)^2}{(a-b)}\) < 0 . Sufficient.

(2) 1 < |a - b|
case 1 : a-b < -1 or a<b-1 --> a<b
case 2: a-b>1 or a> b+1------> a>b
not sufficient.


Ans : A

Author:  EgmatQuantExpert [ 08 May 2015, 00:40 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Going through the solutions posted above, I realized that this question is a good illustration of the importance of first analyzing the question statement before moving to St. 1 and 2 in a DS question.

Most students dived straight into Statement 1 and then tried to draw inferences from St. 1 to determine whether the inequality given in the question statement was true or not.

Imagine if you had done this instead before going to St. 1:

The question is asking if:
\(\frac{1}{a-b} < b-a\)

Case 1: a - b is positive (that is, a > b)

Multiplying both sides of an inequality with with the positive number (a-b) will not change the sign of inequality.

So, the question simplifies to: Is 1 < (b-a)(a-b)

Now, b - a will be negative.

So, the question simplifies to: Is \(1 < -(a-b)^2\) . . . (1)

\((a-b)^2\), being a square term, will be >0 (Note: a - b cannot be equal to zero because then the fraction given in the question: 1/a-b becomes undefined)

So, \(-(a-b)^2\) will be < 0

Therefore, the question simplifies to: Is 1 < (a negative number?)

And the answer is NO.

Case 2: a - b is negative (that is, a < b)

Multiplying both sides of an inequality with the negative number (a-b) will change the sign of inequality.

So, the question simplifies to: Is 1 > (b-a)(a-b)

Now, b - a will be positive.

So, the question simplifies to: Is \(1 > -(a-b)^2\) . . . (2)

Again, by the same logic as above, we see that the question simplifies to: Is 1 > (a negative number?)

And the answer is YES

Thus, from the question statement itself, we've inferred that:

If a > b, the answer to the question asked is NO
If a < b, the answer to the question asked is YES

Thus, the only thing we need to find now is whether a > b or a < b.

Please note how we are going to Statement 1 now with a much simpler 'To Find' task now. :-D

One look at St. 1 and we know that it will be sufficient.

One look at St. 2 and we know that it doesn't give us a clear idea of which is greater between a and b, and so, is not sufficient.

To sum up this discussion, spending time on analyzing the question statement before going to the two statements usually simplifies DS questions a good deal.

Hope this helped!

Japinder

Author:  AbdurRakib [ 11 Apr 2016, 00:00 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

[quote="Bunuel"]SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be YES if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be NO. Not sufficient.


Dear,

Please Clarify me Rewrite like this one.I thought rewite variable without considering its Sign is Wrong

Author:  Bunuel [ 11 Apr 2016, 00:03 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

AbdurRakib wrote:
Bunuel wrote:
SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be YES if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be NO. Not sufficient.


Dear,

Please Clarify me Rewrite like this one.I thought rewite variable without considering its Sign is Wrong


We cannot multiply an inequality by a variable if don't know its sign but we can add/subtract a value to both sides. To get a-b<0 from a < b we are subtracting b from both sides and to get b - a > 0 we are subtracting a from both sides.

Author:  CrackverbalGMAT [ 03 Jun 2019, 10:25 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Let us look at this question differently here:

While solving DS inequality questions, the best approach is to always breakdown the question stem if possible. To breakdown the question stem, there are three hygiene factors, that if followed will simplify your analysis of the question.

1. Always keep the RHS of the inequality as 0
2. Simplify the LHS to a product or division of values (product and division of terms are easier to analyze)
3. Always try and maintain even powered terms (as the sign of them will always be 0 or positive)


Let us breakdown the question stem here:

1/(a - b) > b - a
(1/(a - b)) - (b - a) > 0

Since we have an option to get a squared term, before we take the LCM let us change b - a to a - b.

(1/(a - b)) + (a - b) > 0 ----> (1 + (a - b)^2)/(a - b) > 0

Now since we have deconstructed the question stem to a division we just need to worry about the signs of the numerator and denominator.

The RHS here is > 0, so we the numerator and denominator have to have the same sign. The numerator however will always be positive, since 1 + (a - b)^2 will always be positive. So we just need need the denominator to also be positive.

The entire question stem can now be rephrased as 'Is a - b > 0'?

Statement 1 : a - b < 0

This gives us a definite NO. So sufficient.

Statement 2 : 1 < |a - b|

|a - b| > 1. This only tells us that a - b > 1 or a - b < -1. So a - b can be both positive or negative. Insufficient.

Answer: A

Hope this helps!

Aditya
CrackVerbal Academic Team

Author:  Kinshook [ 31 Aug 2019, 09:13 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Bunuel wrote:
Is 1/(a - b) > b - a ?

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


The Official Guide For GMAT® Quantitative Review, 2ND Edition


Asked: Is 1/(a - b) > b - a ?
1/(a-b) + (a-b) >0
(a-b)^2 + 1 / (a-b) >0
Since numerator >0
Q. a -b >0

(1) a < b
a-b<0
a-b is NOT >0
SUFFICIENT

(2) 1 < |a - b|
a-b > 1 or a-b < -1
NOT SUFFICIENT

IMO A

Author:  siddharth287 [ 29 Jan 2022, 07:02 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Bunuel wrote:
SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be YES if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be NO. Not sufficient.

Answer: A.



statement 1 says a<b so if we substitute values for a and b.
example a = 2 and b = 3
then from original equation we get
1/(2-3) > 3-2 ie 1/-1 > 1 ie -1>1 how is that possible? And why is statment 1 sufficient then

Author:  Bunuel [ 29 Jan 2022, 08:06 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

siddharth287 wrote:
Bunuel wrote:
SOLUTION

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

(1) a < b --> we can rewrite this as: \(a-b<0\) so LHS is negative, also we can rewrite it as: \(b-a>0\) so RHS is positive --> negative<positive. Sufficient.

(2) 1 < |a - b| --> if \(a-b=2\) (or which is the same \(b-a=-2\)) then LHS>0 and RHS<0 and in this case the answer will be YES if \(a-b=-2\) (or which is the same \(b-a=2\)) then LHS<0 and RHS>0 and in this case the answer will be NO. Not sufficient.

Answer: A.



statement 1 says a<b so if we substitute values for a and b.
example a = 2 and b = 3
then from original equation we get
1/(2-3) > 3-2 ie 1/-1 > 1 ie -1>1 how is that possible? And why is statment 1 sufficient then


The question asks whether 1/(a - b) > b - a. If we can give a definite YES answer to the question (YES, 1/(a - b) IS greater than b - a) OR a definite NO answer to the question (No, 1/(a - b) is NOT greater than b - a), then the statement is sufficient (recall that a definite NO answer to a DS question is also sufficient).

Now, since a < b, then a - b < 0 and b - a > 0, thus 1/(a - b) = 1/negative = negative, while b - a = positive, so 1/(a - b) < b - a, which means that the answer to the question is NO, 1/(a - b) is NOT greater than b - a. That's why the first statement is sufficient.

Check the links below for DS Strategies and Tactics




For other subject check Ultimate GMAT Quantitative Megathread.

Hope it helps.

Author:  DHRJ0032 [ 29 Jan 2022, 10:16 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Well this question can be solved very easily with the graphical approach.

Let, a-b=x
Now, rephrase the question:

Is 1/x>-x?
Clearly from graph , for x>0,
1/x>-x.

1)x<0
No, for x<0 , 1/x<-x
SUFFICIENT.

2)|x|>1
For x>1 , YES
for x<-1, NO
INSUFFICIENT.

A it is.

Posted from my mobile device

Attachments:
File comment: Graphical approach
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Author:  sacharya [ 31 Mar 2022, 21:42 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

\(1/(a - b) > b - a ?\)
Simplifying this further
\(1/(a - b) - (b-a) >0 ?\)
\(1+(a-b)^{2}/(a-b) > 0 ?\)

\(1+(a-b)^{2}\) is always positive; we need to find if (a-b) is positive or neg
Stmt 1:
a<b
a-b<0
(a-b) negative
Sufficient

Stmt 2:
1 < |a - b| i.e. |a - b| > 1
(a-b) > 1 OR -(a-b) > 1
case 1. (a-b) > 1, (a-b) will be positive
case 2. -(a-b) > 1
(a-b) < -1 , (a-b) will be negative
Not Sufficient

Ans A

Author:  gmatprepguy2049 [ 11 Apr 2022, 05:34 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

Where are the options for the question? I'm not able to see them

Author:  Bunuel [ 11 Apr 2022, 07:50 ]
Post subject:  Re: Is 1/(a - b) > b - a ? (1) a < b (2) 1 < |a - b|

gmatprepguy2049 wrote:
Where are the options for the question? I'm not able to see them


Hi,

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post ALL YOU NEED FOR QUANT.

Hope this helps.

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