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

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Bunuel
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If |a + b| - c < d, which of the following must be true? [#permalink] New post   05 Oct 2021, 05:30
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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
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D
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Re: If |a + b| - c < d, which of the following must be true? [#permalink] New post   05 Oct 2021, 06:47
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If \(|a+b|-c < d\), which of the following must be true?

\(|a+b|-c < d\)

\(|a+b|<c+ d\), so c+d>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
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If |a + b| - c < d, which of the following must be true? [#permalink] New post   05 Oct 2021, 06:57
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Bunuel wrote:
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


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
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Re: If |a + b| - c < d, which of the following must be true? [#permalink] New post   05 Oct 2021, 07:06
Bunuel wrote:
If \(|a+b|-c < d\), which of the following must be true?

A. I only
B. II only
C. III only
D. II and III only
E. I, II, and III


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
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Re: If |a + b| - c < d, which of the following must be true? [#permalink] New post   03 Nov 2021, 07:54
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Bunuel wrote:
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





This question is a part of Are You Up For the Challenge: 700 Level Questions collection.
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Re: If |a + b| - c < d, which of the following must be true? [#permalink] New post   17 Feb 2023, 00:17
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Bunuel wrote:
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

Solution:

  • We are given \(|a+b|-c < d\)
    \(⇒|a+b|<d+c\)
  • From this we can infer that \(d+c>0\)

    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
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If |a + b| - c < d, which of the following must be true? [#permalink] New post   Updated on: 18 Mar 2023, 03:52
SaquibHGMATWhiz wrote:

  • We are given \(|a+b|-c < d\)
    \(⇒|a+b|<d+c\)
  • From this we can infer that \(d+c>0\)

[/color]



I have difficulty seeing, why |a+b|<d+c must mean, that d+c > 0 ... can't a and b for example both be 0?

Thanks!

Edit: I just realized how stupid this question is :lol: sorry & thank you for the answer anyway!

Originally posted by mliv on 18 Mar 2023, 03:28.
Last edited by mliv on 18 Mar 2023, 03:52, edited 1 time in total.
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If |a + b| - c < d, which of the following must be true? [#permalink] New post   18 Mar 2023, 03:36
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mliv wrote:
SaquibHGMATWhiz wrote:

  • We are given \(|a+b|-c < d\)
    \(⇒|a+b|<d+c\)
  • From this we can infer that \(d+c>0\)

[/color]



I have difficulty seeing, why |a+b|<d+c must mean, that d+c > 0 ... can't a and b for example both be 0?


Thanks!


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.
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Re: If |a + b| - c < d, which of the following must be true? [#permalink] New post   18 Mar 2023, 06:15
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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Re: If |a + b| - c < d, which of the following must be true? [#permalink]
18 Mar 2023, 06:15
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