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Bunuel
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Is 1/(a + b) > b - a? 23 Mar 2017, 06:07
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Is 1/(a + b) > b - a?

(1) a + b > 0
(2) b^2 – a^2 > 1
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C
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CrackverbalGMAT
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Is 1/(a + b) > b - a? 24 Mar 2017, 05:07
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Hi,

The following points will hold you in good stead while solving questions on inequalities.

1. Always break down the question stem
2. Keep the right hand side of the inequality as 0 (This helps in an analysis and will save you the tedious task of plugging in values)

The question here is 'Is 1/(a + b) > b - a?'. Let us break down the question stem by keeping the right hand side 0 and simplifying.

1/(a + b) - (b - a) > 0
Simplifying the left hand side we get
(1 - (b - a)(b + a))/(a + b) > 0 ------> (1 - (b^2 - a^2)/(a + b)) > 0

So now the question can be rephrased as 'Is (1 - (b^2 - a^2)/(a + b)) > 0'

Statement 1 : a + b > 0

Here the denominator a + b in (1 - (b^2 - a^2)/(a + b)) is positive, but we have no information whether the numerator is positive or negative. So it is possible for us to get a YES and a NO. Insufficient.

Statement 2 : b^2 – a^2 > 1

The numerator 1 - (b^2 - a^2) will always be negative since b^2 - a^2 is greater than 1, but we have no information about the denominator a + b. Insufficient.

Combining 1 and 2 :

We know that the numerator is negative and the denominator is positive, so the entire term (1 - (b^2 - a^2)/(a + b)) will always be negative. This gives us a definite NO. Sufficient.

Hope this helps!

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Is 1/(a + b) > b - a? 23 Mar 2017, 10:14
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False. Because a+b>0, 1/(a+b) will ALWAYS be less than 1. b^2-a^2>1 can be square rooted to b-a>1. This means b-a>1/(a+b). Always.

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Is 1/(a + b) > b - a? 23 Mar 2017, 10:31
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C.

Statement 1

a+b > 0

Let a=b=1 => (1/2 > 0) => Yes
Lets a=0 & b=1 => (1 > 1 ) => No

Thus, the statement is insufficient.


Statement 2

b^2 - a^2 > 1

Let a=1 & b=5 => (1/6 > 4) => No
Let a=0 & b=-2 => (1/-2 > -2) => Yes

Thus, statement 2 is also insufficient.


Combining both the statements:

We know that b has to be the bigger value since b^2 > 1 + a^2 ( which is always a +ve value )
Thus, a+b will also have be +ve ( since we cannot have a negative value for b )

Therefore, we can consider the above example where a=1 & b=5 we get a definite NO.
Hence, the answer is C.


Hope this reasoning is correct!
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Is 1/(a + b) > b - a? Updated on: 23 Mar 2017, 13:42
Originally posted by LeveragedTiger on 23 Mar 2017, 12:37.
Last edited by LeveragedTiger on 23 Mar 2017, 13:42, edited 2 times in total.
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Original statement:

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

This can be re-arranged as 1 > (b - a)(b + a) if (a + b) is positive, or 1 < (b - a)(b + a) if (a + b) is negative.

Or, is 1 > (b^2 - a^2) when (a + b) is positive, or is 1 < (b^2 - a^2) if (a + b) is negative.

I:

(a + b) is positive. We therefore know we need to answer the inequality 1 > (b^2 - a^2). But we don't know anything about a, b, or (b^2 - a^2). Insufficient.

II:

b^2 - a^2 > 1.

On first glance, this looks like it satisfies the re-arrangement of the original statement. However, we do not know if (a + b) is positive or negative, so we do not know if the answer to the question is Yes or No. If (a + b) is positive, the answer is no, If (a + b) is negative, the answer is yes. Insufficient.

Combine:

We know that (a + b) is positive, and that b^2 - a^2 > 1.

From the original statement, we are asking if 1 > b^2 - a^2 if (a + b) is positive.

Since (a + b) is positive, b^2 - a^2 > 1, and the answer is No. Sufficient.
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Is 1/(a + b) > b - a? 23 Mar 2017, 12:41
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Hi Bunuel,

I somehow got this ques correct, but is there a better way to solve this rather than putting the values?
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Is 1/(a + b) > b - a? 25 Jul 2025, 13:38
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One question, for evaluating both at the same time, for example:

if b=1/2 and a=1/4, it satisfy that b>a and that a+b>0.

Solving, you have 1/(1/2+1/4)>1/2-1/4; that's 1/(3/4)>1/4 = 4/3>1/4. So, the solution to the question is YES.

At the same time, if we try b=5 and a=1; that's 1/6<6. So, that's a NO.

Why is not E? Thanks,
CrackverbalGMAT
Hi,

The following points will hold you in good stead while solving questions on inequalities.

1. Always break down the question stem
2. Keep the right hand side of the inequality as 0 (This helps in an analysis and will save you the tedious task of plugging in values)

The question here is 'Is 1/(a + b) > b - a?'. Let us break down the question stem by keeping the right hand side 0 and simplifying.

1/(a + b) - (b - a) > 0
Simplifying the left hand side we get
(1 - (b - a)(b + a))/(a + b) > 0 ------> (1 - (b^2 - a^2)/(a + b)) > 0

So now the question can be rephrased as 'Is (1 - (b^2 - a^2)/(a + b)) > 0'

Statement 1 : a + b > 0

Here the denominator a + b in (1 - (b^2 - a^2)/(a + b)) is positive, but we have no information whether the numerator is positive or negative. So it is possible for us to get a YES and a NO. Insufficient.

Statement 2 : b^2 – a^2 > 1

The numerator 1 - (b^2 - a^2) will always be negative since b^2 - a^2 is greater than 1, but we have no information about the denominator a + b. Insufficient.

Combining 1 and 2 :

We know that the numerator is negative and the denominator is positive, so the entire term (1 - (b^2 - a^2)/(a + b)) will always be negative. This gives us a definite NO. Sufficient.

Hope this helps!

CrackVerbal Academics Team
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Is 1/(a + b) > b - a? Updated on: 25 Jul 2025, 14:45
Originally posted by Bunuel on 25 Jul 2025, 14:09.
Last edited by Bunuel on 25 Jul 2025, 14:45, edited 1 time in total.
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aaronmart14
Is 1/(a + b) > b - a?

(1) a + b > 0
(2) b^2 – a^2 > 1


One question, for evaluating both at the same time, for example:

if b=1/2 and a=1/4, it satisfy that b>a and that a+b>0.

Solving, you have 1/(1/2+1/4)>1/2-1/4; that's 1/(3/4)>1/4 = 4/3>1/4. So, the solution to the question is YES.

At the same time, if we try b=5 and a=1; that's 1/6<6. So, that's a NO.

Why is not E? Thanks,
CrackverbalGMAT
Hi,

The following points will hold you in good stead while solving questions on inequalities.

1. Always break down the question stem
2. Keep the right hand side of the inequality as 0 (This helps in an analysis and will save you the tedious task of plugging in values)

The question here is 'Is 1/(a + b) > b - a?'. Let us break down the question stem by keeping the right hand side 0 and simplifying.

1/(a + b) - (b - a) > 0
Simplifying the left hand side we get
(1 - (b - a)(b + a))/(a + b) > 0 ------> (1 - (b^2 - a^2)/(a + b)) > 0

So now the question can be rephrased as 'Is (1 - (b^2 - a^2)/(a + b)) > 0'

Statement 1 : a + b > 0

Here the denominator a + b in (1 - (b^2 - a^2)/(a + b)) is positive, but we have no information whether the numerator is positive or negative. So it is possible for us to get a YES and a NO. Insufficient.

Statement 2 : b^2 – a^2 > 1

The numerator 1 - (b^2 - a^2) will always be negative since b^2 - a^2 is greater than 1, but we have no information about the denominator a + b. Insufficient.

Combining 1 and 2 :

We know that the numerator is negative and the denominator is positive, so the entire term (1 - (b^2 - a^2)/(a + b)) will always be negative. This gives us a definite NO. Sufficient.

Hope this helps!

CrackVerbal Academics Team
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b = 1/2 and a = 1/4 do not satisfy b^2 − a^2 > 1. Every pair that satisfies both a + b > 0 and b^2 − a^2 > 1 will give a NO to the question "Is 1/(a + b) > b - a?" So the correct answer is C, not E.

P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

  • Word Problems
  • Work Problems
  • Distance Problems
  • Mixture Problems
  • Percent and Interest Problems
  • Overlapping Sets Problems
  • Statistics Problems
  • Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.­
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Is 1/(a + b) > b - a? Updated on: 25 Jul 2025, 14:46
Originally posted by aaronmart14 on 25 Jul 2025, 14:30.
Last edited by Bunuel on 25 Jul 2025, 14:46, edited 1 time in total.
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Thank you very much Bunuel!!
Bunuel
aaronmart14
Is 1/(a + b) > b - a?

(1) a + b > 0
(2) b^2 – a^2 > 1


One question, for evaluating both at the same time, for example:

if b=1/2 and a=1/4, it satisfy that b>a and that a+b>0.

Solving, you have 1/(1/2+1/4)>1/2-1/4; that's 1/(3/4)>1/4 = 4/3>1/4. So, the solution to the question is YES.

At the same time, if we try b=5 and a=1; that's 1/6<6. So, that's a NO.

Why is not E? Thanks,
CrackverbalGMAT
Hi,

The following points will hold you in good stead while solving questions on inequalities.

1. Always break down the question stem
2. Keep the right hand side of the inequality as 0 (This helps in an analysis and will save you the tedious task of plugging in values)

The question here is 'Is 1/(a + b) > b - a?'. Let us break down the question stem by keeping the right hand side 0 and simplifying.

1/(a + b) - (b - a) > 0
Simplifying the left hand side we get
(1 - (b - a)(b + a))/(a + b) > 0 ------> (1 - (b^2 - a^2)/(a + b)) > 0

So now the question can be rephrased as 'Is (1 - (b^2 - a^2)/(a + b)) > 0'

Statement 1 : a + b > 0

Here the denominator a + b in (1 - (b^2 - a^2)/(a + b)) is positive, but we have no information whether the numerator is positive or negative. So it is possible for us to get a YES and a NO. Insufficient.

Statement 2 : b^2 – a^2 > 1

The numerator 1 - (b^2 - a^2) will always be negative since b^2 - a^2 is greater than 1, but we have no information about the denominator a + b. Insufficient.

Combining 1 and 2 :

We know that the numerator is negative and the denominator is positive, so the entire term (1 - (b^2 - a^2)/(a + b)) will always be negative. This gives us a definite NO. Sufficient.

Hope this helps!

CrackVerbal Academics Team

b = 1/2 and a = 1/4 do not satisfy b^2 − a^2 > 1. Every pair that satisfies both a + b > 0 and b^2 − a^2 > 1 will give a NO to the question "Is 1/(a + b) > b - a?" So the correct answer is C, not E.

P.S. Pure algebraic questions are no longer a part of the DS syllabus of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

  • Word Problems
  • Work Problems
  • Distance Problems
  • Mixture Problems
  • Percent and Interest Problems
  • Overlapping Sets Problems
  • Statistics Problems
  • Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check GMAT Syllabus for Focus Edition

You can also visit the Data Sufficiency forum and filter questions by OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.­
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