Is |a - b|

Question

Is  |a - b| < |a| +|b| ?

(1)  \frac{a}{b}<0

(2)  a^2b<0

mikemcgarry · Answer

Dear DH99 , I'm happy to respond. :-)

This is a very clever question. Let's think about the prompt. If a & b have the same sign, then subtracting produces a difference of a smaller absolute value. 5 - 3 = 2 (-5) - (-3) = -2 These are examples of choices that would produce "yes" answers for the prompt question. If a & b have opposite signs, the subtracting produces a difference that has the same absolute value as the sum of the absolute values of a & b separately. 5 - (-3) = 8 and |8| = 8 = |5| + |-3| (-5) - 3 = -8 and |-8| = 8 = |-5| + |3| These are examples of choices that would produce "no" answers for the prompt question. So really, the prompt question is: do a and b have the same sign? Statement #1 : \frac{a}{b} <0 When is a fraction negative? It's negative when the numerator and denominator have opposite signs. Thus, a & b have opposite signs. We can give a definitive "no" to the prompt question. Because we can give a definitive answer, we know that Statement #1, alone and by itself, must be sufficient . Statement #2 : a^2b <0 I assume that this was entered correctly, and that DH99 meant a^2b and not a^{2b} . Taking this statement at face value, we know a^2 is always positive, regardless of the sign of a, so we know that b just be negative. Thus, we know the sign of b, but the sign of a could be anything. We are not able to give a definitive answer to the prompt. Thus, we know that Statement #2, alone and by itself, must be insufficient . OA = (A) Does all this make sense? Mike :-)

DHAR · Answer

Yes,thanks  mike

sahilvijay · Answer

Clearly from A we are getting definite NO. 
See the attached pic.

niks18 · Answer

Statement 1: implies that either a<0 or b<0 Putting the values of a & b in the inequality as per the scenario if a<0 , then |a-b| = |-a-b| = |a+b| = |a| +|b| . so we get a NO for the question stem if b<0 , then |a-b| = |a-(-b)| = |a+b| = |a| + |b| . so we get a NO for the question stem Hence Sufficient Statement 2:/b] implies that b<0 but a<0 or a>0 Putting the values of a & b in the inequality as per the scenario if both a<0 and b<0 , then |a-b| = |-a-(-b)| = |-a+b|<|a| + |b| . so we get a YES for the question stem but if a>0 and b<0 , then |a-b| = |a-(-b)| = |a+b| = |a| + |b| . so we get a NO for the question stem Hence Insufficient Option A

hellosanthosh2k2 · Answer

|a-b| < |a| + |b| ? squaring on both sides, (as LHS > 0, RHS > 0, it is safe to square) => a^2 + b^2 - 2ab < a^2 + b^2 + 2|a||b| ? => -2ab < 2|a||b| ? => -ab < |ab| ? => question is reduced to ab > 0 ? => a and b are of same sign? Let us attack the statements Statement 1: a/b < 0 => which means a and b are of opposite sign, sufficient to answer the question as "NO" Statement 2: a^2 * b < 0 => b is negative, but we don't about the sign of a => InSufficient Answer (A)

Kinshook · Answer

Given: Is |a-b| < |a| +|b| ?
This is true only if a & b are of same sign

Statement 1:  \frac{a}{b} <0
a & b are of different signs.
a & b are NOT of same signs
SUFFICIENT

Statement 2:  a^2b <0
b<0 and  a 
eq 0 
Since sign of a is not known
NOT SUFFICIENT

IMO A

Is |a - b| < |a| + |b|? (1) a/b <0 (2) a^2b < 0

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