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# If x + y + z > 0, is z > 1?

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Math Expert
Joined: 02 Sep 2009
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If x + y + z > 0, is z > 1? [#permalink]

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12 Feb 2014, 01:11
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The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 < 0

Data Sufficiency
Question: 89
Category: Algebra Inequalities
Page: 158
Difficulty: 650

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Re: If x + y + z > 0, is z > 1? [#permalink]

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12 Feb 2014, 01:12
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SOLUTION

Note that:
You can only add inequalities when their signs are in the same direction:

If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.

You can only apply subtraction when their signs are in the opposite directions:

If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.

Back to the original question:

If x + y + z > 0, is z > 1?

(1) z > x + y + 1 --> as the signs are in the same direction we can add two inequalities: $$(x+y+z)+z>x+y+1$$ --> $$2z>1$$ --> $$z>\frac{1}{2}$$, so z may or may not be more than 1. Not sufficient.

(2) x + y + 1 < 0 --> as the signs are in the opposite direction we can subtract two inequalities: $$(x+y+z)-(x+y+1)>0$$ --> $$z-1>0$$ --> $$z>1$$. Sufficient.

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Re: If x + y + z > 0, is z > 1? [#permalink]

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12 Feb 2014, 11:02
If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 > 0

Sol: st1 says z>x+y+1 and x+y+z>0

If x=-2, y=-3,z=6 then z>1
But if x=-0.1,y=-0.1 thus z>0.8 so it may or may not be greater than 1
Not sufficient. A and D ruled out

St 2 says x+y+1>0
Again x=-1/2,y=1/2 then z can take any value greater than or less than 1 but if x=1000,y=1 then again z can take any value so clearly
Not sufficient

On combining we get that z>0 but z may or may not be greater than1

Ans E

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Re: If x + y + z > 0, is z > 1? [#permalink]

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12 Feb 2014, 16:47
If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 > 0

Sol: st1 says z>x+y+1 and x+y+z>0

Using stmt 1 : z-x-y>1, therefore z will be >1.

Therefore stmt A is sufficient.
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Joined: 29 Oct 2013
Posts: 15
Re: If x + y + z > 0, is z > 1? [#permalink]

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12 Feb 2014, 19:32
X017in wrote:
If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 > 0

Sol: st1 says z>x+y+1 and x+y+z>0

Using stmt 1 : z-x-y>1, therefore z will be >1. --->x, y can be negative then conclusion fails

Therefore stmt A is sufficient.

Ans-A is not correct I guess
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GMAT 2: 720 Q47 V41
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Re: If x + y + z > 0, is z > 1? [#permalink]

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13 Feb 2014, 07:04
2
KUDOS
syamen wrote:
X017in wrote:
If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 > 0

Sol: st1 says z>x+y+1 and x+y+z>0

Using stmt 1 : z-x-y>1, therefore z will be >1. --->x, y can be negative then conclusion fails

Therefore stmt A is sufficient.

Ans-A is not correct I guess

OK. Taking a different approach :

x + y + z > 0
- x + y + 1 > 0
= z-1>0
= z>1

So B is sufficient?
Intern
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Posts: 25
Re: If x + y + z > 0, is z > 1? [#permalink]

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13 Feb 2014, 09:40
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Re: If x + y + z > 0, is z > 1? [#permalink]

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13 Feb 2014, 16:08
Stmt 1 : z > x + y + 1 i.e x+y-z > -1
Is z>1 ?
yes, put z = 2 & x+y =3 so x+y - z =1, which is > -1 ... also we maintain the given condition x+y+z>0
no, put z = 0 & x+y =3 so x+y - z =3 , which is > -1 again we pick nos such that x+y+z> is not violated

A is insuff ..so A and D eliminated

Stmt 2 : x + y + 1 > 0 i.e. x+y > -1
Is z> 1 ?
yes i.e pick any value for x+y >-1 so that x+y+z>0 is not violated .. So at the least x+y =-1 , then z has to be z > 1 for x+y + z > 0..

no, z = -2 and x+y = 5 then x+y+z = 3 i.e. > 0

B is insuff

So OA is E
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Re: If x + y + z > 0, is z > 1? [#permalink]

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13 Feb 2014, 22:14
1
KUDOS
Expert's post
X017in wrote:
syamen wrote:
X017in wrote:
If x + y + z > 0, is z > 1?

(1) z > x + y + 1
(2) x + y + 1 > 0

Sol: st1 says z>x+y+1 and x+y+z>0

Using stmt 1 : z-x-y>1, therefore z will be >1. --->x, y can be negative then conclusion fails

Therefore stmt A is sufficient.

Ans-A is not correct I guess

OK. Taking a different approach :

x + y + z > 0
- x + y + 1 > 0
= z-1>0
= z>1

So B is sufficient?

You cannot subtract inequalities.

Consider:

4>2 ....(i)
3>-2 ...(ii)

(i) - (ii) gives
1 > 4 (not true)

You can add them if they have the same inequality sign. If they have opposite signs, multiply one of them by -1, flip the inequality sign and then add them.
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Math Expert
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Posts: 43834
Re: If x + y + z > 0, is z > 1? [#permalink]

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17 Feb 2014, 01:14
2
KUDOS
Expert's post
SOLUTION

Note that:
You can only add inequalities when their signs are in the same direction:

If $$a>b$$ and $$c>d$$ (signs in same direction: $$>$$ and $$>$$) --> $$a+c>b+d$$.
Example: $$3<4$$ and $$2<5$$ --> $$3+2<4+5$$.

You can only apply subtraction when their signs are in the opposite directions:

If $$a>b$$ and $$c<d$$ (signs in opposite direction: $$>$$ and $$<$$) --> $$a-c>b-d$$ (take the sign of the inequality you subtract from).
Example: $$3<4$$ and $$5>1$$ --> $$3-5<4-1$$.

Back to the original question:

If x + y + z > 0, is z > 1?

(1) z > x + y + 1 --> as the signs are in the same direction we can add two inequalities: $$(x+y+z)+z>x+y+1$$ --> $$2z>1$$ --> $$z>\frac{1}{2}$$, so z may or may not be more than 1. Not sufficient.

(2) x + y + 1 < 0 --> as the signs are in the opposite direction we can subtract two inequalities: $$(x+y+z)-(x+y+1)>0$$ --> $$z-1>0$$ --> $$z>1$$. Sufficient.

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Re: If x + y + z > 0, is z > 1? [#permalink]

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23 Mar 2016, 10:01
Statement II is quite obvious.

If x + y + z > 0
(2) x + y + 1 < 0

We are initially told that x + y + z > 0, but statement II tells us that x + y + 1 < 0. Hence, given the question prompt, z must be greater than 1.
Manager
Joined: 17 Aug 2015
Posts: 105
Re: If x + y + z > 0, is z > 1? [#permalink]

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03 May 2016, 21:02
My approach to this problem
st #1-- z>x+y+1
Take x = -3 , y= -4 then this implies that z> -6 . NS y

st#2 -- x+y+1<0
ok. from the prompt x+y+z>0 , therefore x+y+z > x+y+1 . This is since postive is always greater than negative.
Then we can take away x and y from both sides by adding -x and -y to both sides one by one.
what remains is z>1 . Sufficient
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Posts: 44
Re: If x + y + z > 0, is z > 1? [#permalink]

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30 Jun 2016, 04:32
Given x + y + z > 0
(1) says z > x + y + 1

x + y + 2z > x + y + 1
z > 0.5

So, we can't say for sure that z > 1. Not sufficient.

(2) x + y + 1 < 0
x + y < -1

Multiplying both sides by -1
-x - y > 1

Question says: x + y + z > 0

z > 1
So, (2) is sufficient.
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Re: If x + y + z > 0, is z > 1? [#permalink]

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25 Aug 2017, 12:09
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Re: If x + y + z > 0, is z > 1?   [#permalink] 25 Aug 2017, 12:09
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