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

Question

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

eunjun · Answer

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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ashwink · Answer

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!

LeveragedTiger · Answer

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.

AR15J · Answer

Hi Bunuel,

I somehow got this ques correct, but is there a better way to solve this rather than putting the values?

CrackverbalGMAT · Answer

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

aaronmart14 · Answer

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,

Bunuel · Answer

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.

aaronmart14 · Answer

Thank you very much Bunuel!!

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

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