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# what are solutions fora if |a+|a|||a|a|| = 0

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Director
Joined: 26 Feb 2006
Posts: 900
what are solutions fora if |a+|a|||a|a|| = 0 [#permalink]

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17 May 2007, 21:49
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what are solutions fora if |a+|a||–|a–|a|| = 0 ?
SVP
Joined: 01 May 2006
Posts: 1796

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17 May 2007, 23:26
To me, a=0 is the only solution ....Let check it now

o If a > 0, then
|a+|a||–|a–|a|| = 0
<=> |a+a| - |a-a| = 0 as a > 0
<=> 2*|a| = 0
<=> 2*a = 0 as a > 0
<=> a = 0.... No solution because a must be positive.

o If a =< 0, then
|a+|a||–|a–|a|| = 0
<=> |a+(-a)| - |a-(-a)| = 0 as a =< 0
<=> -2*|a| = 0
<=> -2*(-a) = 0 as a =< 0
<=> a = 0.... 1 Solution as a can be 0 or negative

So, the solution is:
a=0
GMAT Club Legend
Joined: 07 Jul 2004
Posts: 5042
Location: Singapore

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01 Oct 2007, 18:41
I see this as |a+|a|| = |a-|a||

Positive case:
a+|a| = a - |a|
|a| = -|a|

Since |a| is always positive, so a must be 0.

Negative case:
a+|a| = -a + |a|
a = -a

Only 0 fits the bill.

So a has only 1 solution, that is 0.
01 Oct 2007, 18:41
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# what are solutions fora if |a+|a|||a|a|| = 0

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