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Which of the following inequalities is always true for any

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Which of the following inequalities is always true for any [#permalink] New post 09 Nov 2012, 11:03
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Which of the following inequalities is always true for any real number 'a' and 'b'?

(A) |a + b| = |a| + |b|
(B) |a + b| > |a| + |b|
(C) |a + b| <= |a| + |b|
(D) |a - b| <= |a| - |b|
(E) |a - b| > |a| - |b|
[Reveal] Spoiler: OA
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Re: Which of the following inequalities is always true for any [#permalink] New post 09 Nov 2012, 12:32
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It seems such an easy answer, but I would go with (C). No thinking here bro, you got to know one of the fundamental properties of absolute value, which is subadditivity in a nutshell means that sum of two any elements is something (some number) which is less than or equal to the sum of the each element taken on itself separately, i.e. |a + b| <= |a| + |b|

Please, correct me if I went awry
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Re: Which of the following inequalities is always true for any [#permalink] New post 10 Nov 2012, 05:04
derekgmat wrote:
Which of the following inequalities is always true for any real number 'a' and 'b'?

(A) |a + b| = |a| + |b|
(B) |a + b| > |a| + |b|
(C) |a + b| <= |a| + |b|
(D) |a - b| <= |a| - |b|
(E) |a - b| > |a| - |b|


Property worth remembering:

1. Always true: |x+y|\leq{|x|+|y|}, note that "=" sign holds for xy\geq{0} (or simply when x and y have the same sign);

2. Always true: |x-y|\geq{|x|-|y|}, note that "=" sign holds for xy>{0} (so when x and y have the same sign) and |x|>|y| (simultaneously).

Hope it helps.
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Re: Which of the following inequalities is always true for any [#permalink] New post 20 Nov 2012, 10:10
Please refer this brilliant resource (it includes the above mentioned properties and other relevant information)

gmat-math-book-in-downloadable-pdf-format-130609.html
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Re: Which of the following inequalities is always true for any [#permalink] New post 06 Dec 2012, 01:17
:wink:

One must memorize and understand this property by heart.
|x| + |y| >= |x+y|
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Re: Which of the following inequalities is always true for any [#permalink] New post 07 Jan 2013, 08:21
What is the prove for this properties? Can someone prove this? Pleaseee
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Re: Which of the following inequalities is always true for any [#permalink] New post 08 Jan 2013, 03:57
KevinBrink wrote:
What is the prove for this properties? Can someone prove this? Pleaseee


Square |x+y|\leq{|x|+|y|} (we can safely do that since both sides are non-negative):

x^2+2xy+y^2\leq{x^2+2|xy|+y^2} --> 2xy\leq{2|xy|} --> xy\leq{|xy|} --> always true.

The same way for the second inequality.
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Re: Which of the following inequalities is always true for any   [#permalink] 08 Jan 2013, 03:57
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