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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]

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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]

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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]

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10 Nov 2012, 05:04
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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]

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08 Jan 2013, 03:57
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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]

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20 Nov 2012, 10:10
Please refer this brilliant resource (it includes the above mentioned properties and other relevant information)

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Re: Which of the following inequalities is always true for any [#permalink]

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06 Dec 2012, 01:17

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]

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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]

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16 Dec 2013, 12:29
Bunuel wrote:
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.

Let me see if Ive interpreted you correctly:

$$|x+y|\leq{|x|+|y|}$$ means that "!x! is always denoted in its positive value, and so is !y!, and thus !x+y!, which can take two values, can never be greater than the addition of the two positive values !x! and !y!"

By the same token:

$$|x-y|\geq{|x|-|y|}$$ means that "since both !x! and !y! in the right hand side are denoted in their positive value, their subtracted value can never be bigger than !x - y!"

Is that interpretation correct? That is how I solved the question.
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Re: Which of the following inequalities is always true for any [#permalink]

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15 Feb 2015, 19:35
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Re: Which of the following inequalities is always true for any [#permalink]

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02 Apr 2016, 01:24
Hello from the GMAT Club BumpBot!

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Re: Which of the following inequalities is always true for any [#permalink]

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13 Apr 2017, 19:17
Hello from the GMAT Club BumpBot!

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Re: Which of the following inequalities is always true for any   [#permalink] 13 Apr 2017, 19:17
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