Which of the following inequalities is always true for any

Question

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|

Bunuel · Accepted Answer

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.

Javoni · Answer

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

Please, correct me if I went awry

greatps24 · Answer

Please refer this brilliant resource (it includes the above mentioned properties and other relevant information) gmat-math-book-in-downloadable-pdf-format-130609.html

mbaiseasy · Answer

One must memorize and understand this property by heart. |x| + |y| >= |x+y|

KevinBrink · Answer

What is the prove for this properties? Can someone prove this? Pleaseee

Bunuel · Answer

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.

aeglorre · Answer

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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Which of the following inequalities is always true for any

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Which of the following inequalities is always true for any

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