If |a + b| - c

Question

If |a+b|-c < d , which of the following must be true? I. a+b > 0 II. d > 0 whenever c < 0 III. d+c > 0 A. I only B. II only C. III only D. II and III only E. I, II, and III

chetan2u · Answer

If |a+b|-c < d , which of the following must be true? |a+b|-c < d |a+b|0 I. a+b > 0 Not necessarily true. II. d > 0 whenever c < 0 We know c+d>0. If c<0, then d>|c|>c>0 Always true III. d+c > 0 We have already proved it. Only II and III

TestPrepUnlimited · Answer

We have  d + c > |a + b| \geq 0 , so we must have d + c > 0.

Therefore III is true. We also cannot have c and d both negative since the sum has to be positive, then II must be true as well.

Ans: D

Crytiocanalyst · Answer

I.  a+b > 0  
a and b can take any values by changing c and d therefore out

II.  d > 0  whenever  c < 0 
modulus is always positive c is negative then minus of minus is positive therefore the eqn holds

III.  d+c > 0 
This can be achived by taking C to the other side therefore true

Therefore IMO D

Bunuel · Answer

This question is a part of  Are You Up For the Challenge: 700 Level Questions  collection.

SaquibHGMATWhiz · Answer

Solution: We are given |a+b|-c < d ⇒|a+b|0 I. a+b > 0 Not necessarily true II. d > 0 whenever c < 0 We know that d+c>0 So, if c < 0 then d has to be greater than 0 or d>0 Thus, this must be true III. d+c > 0 This must be true as inferred from question statement itself Hence the right answer is Option D

mliv · Answer

I have difficulty seeing, why |a+b| 0 ... can't a and b for example both be 0? Thanks! Edit: I just realized how stupid this question is

sorry & thank you for the answer anyway!

Bunuel · Answer

Yes, both a and b can be 0 and in the case |a + b| = 0 and you'd get 0 < d + c. The point is that the absolute value is always non-negative, hence, the smallest possible value for |a + b| is 0. Therefore, from |a + b| < d + c , we get 0 < d + c.

Kinshook · Answer

Asked: If |a+b|-c < d , which of the following must be true? 0 <= |a+b| < c+d I. a+b > 0 MAY OR MAY NOT BE TRUE II. d > 0 whenever c < 0 c + d > 0; If c<0 - > d>0 MUST BE TRUE III. d+c > 0 d + c > |a+b| >= 0 MUST BE TRUE A. I only B. II only C. III only D. II and III only E. I, II, and III IMO D

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If |a + b| - c < d, which of the following must be true?

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