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

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.

Hi Bunuel,

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

hi Engr2012 chetan2u please help on the above question

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