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# If x + y + z < 1, is z < −1?

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Math Expert
Joined: 02 Sep 2009
Posts: 54370
If x + y + z < 1, is z < −1?  [#permalink]

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24 Jul 2017, 23:25
00:00

Difficulty:

55% (hard)

Question Stats:

35% (01:43) correct 65% (01:41) wrong based on 35 sessions

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If $$x + y + z < 1$$, is $$z < −1$$?

(1) x and y are positive numbers.
(2) xy = 1.

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Joined: 27 May 2017
Posts: 12
Re: If x + y + z < 1, is z < −1?  [#permalink]

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Updated on: 25 Jul 2017, 04:35
St 1. if x and y are positive numbers i.e 1 that means x + y = 2. for the sum to be <1 z has to be - 2. so Z < - 1

If X and Y are fractions i.e 1/2 each and z is 1/2 X+Y+Z < 1 NS

St 2. XY = 1. i.e XY = both positive or both negative. if positive then z < - 1. if negative then z can be 0, 1 or 2. NS

Combined Suff Ans C

Sent from my SM-G935F using GMAT Club Forum mobile app

Originally posted by Ejiroosa on 25 Jul 2017, 01:27.
Last edited by Ejiroosa on 25 Jul 2017, 04:35, edited 1 time in total.
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Re: If x + y + z < 1, is z < −1?  [#permalink]

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25 Jul 2017, 01:36
1
Bunuel wrote:
If $$x + y + z < 1$$, is $$z < −1$$?

(1) x and y are positive numbers.
(2) xy = 1.

x + y + z < 1

(1) x & y are positive
Assume x=1/4, y= 1/3, z=1/3 here z is not less than -1
Now Assume x = 1/4, y=1/3, z=-3 here z is less than -1
Hence Not Sufficient

(2) xy = 1
Assume x = -1, y=-2, z=-3, here z is less than -1
Now Assume x=-1, y=-1, z=3/2 here z is not less than -1
Hence not sufficient

On combining
Since x and y are positive and xy=1
x=1, y=1 then z has to be <-1
if x=1/3, y=3 then z has to be <-1
if x = 3/4 y = 4/3 then also z has to be <-1

Also
since xy=1 we can write x = 1/y and then
x + y + z < 1
1/y + y + z < 1
z<1-y-1/y
z<1-(y+1/y) (we know x+1/x>=2 if x>0)
taking minimum value of y+1/y
z<1-2
z<-1
Hence C
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Re: If x + y + z < 1, is z < −1?   [#permalink] 25 Jul 2017, 01:36
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