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# If wz < 2, is z < 1?

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Manager
Joined: 17 Aug 2009
Posts: 204
If wz < 2, is z < 1? [#permalink]

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02 Jan 2010, 17:16
2
1
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Difficulty:

15% (low)

Question Stats:

81% (00:51) correct 19% (01:07) wrong based on 255 sessions

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If wz < 2, is z < 1?

(1) w > 2
(2) z < 2
Math Expert
Joined: 02 Sep 2009
Posts: 46271

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02 Jan 2010, 17:41
1
2
If wz < 2, is z < 1?

(1) w > 2. Subtract $$w > 2$$ from $$wz < 2$$ (we can do it as the signs are in opposite direction):

$$wz-w<2-2$$ --> $$w(z-1)<0$$, as w is positive (given w>2), then the product to be negative $$z-1$$ must be negative --> $$z-1<0$$ --> $$z<1$$. Sufficient.

(2) z < 2. If $$z=1.5>1$$ and $$w=0$$ (wz<2), then the answer is YES but if $$z=0<1$$ and $$w=0$$ (wz<0), then the answer is NO. Not sufficient.

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Manager
Joined: 17 Aug 2009
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02 Jan 2010, 17:46
Thanks Bunuel

Even i got A. However, the OA given in the GMAC paper test is D.

I guess the OA (like many others in the paper tests) is wrong
Intern
Joined: 14 Jul 2013
Posts: 25
Re: If wz < 2, is z < 1? [#permalink]

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14 Apr 2014, 01:31
1
If wz < 2, is z < 1?

(1) w > 2
(2) z < 2

Sol.
opt.1 => w>2
i.e. z(something>2)<2
=> z<2/(something >2)
in the step above, R.H.S will always be <1. Hence Z<1

opt.2 => z<2
from this we can't say whether z will be less then 1.
Hence, A.
BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2642
GRE 1: 323 Q169 V154
Re: If wz < 2, is z < 1? [#permalink]

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13 Mar 2016, 06:01
1
Here the simple way to conclude that statement 1 is sufficient is because w is always positive and greater than 2
for w=2 which it cant be => z=1
so as Z increases z will decrease => Z<1
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Re: If wz < 2, is z < 1? [#permalink]

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09 Apr 2017, 08:03
1
Top Contributor
zaarathelab wrote:
If wz < 2, is z < 1?

(1) w > 2
(2) z < 2

Target question: Is z < 1?

Given: wz < 2

Statement 1: w > 2
This means that w is POSITIVE, which means we can divide both sides of the given inequality, wz < 2, by w.
We get: z < 2/w
First, since w is POSITIVE, we know that 2/w is POSITIVE, which means z is POSITIVE
Second, since w > 2, we know that 2/w will be less than 1, since the denominator is greater than the numerator
So, we can write 2/w < 1
Since z < 2/w, we can write: z < 2/w < 1
This means we can conclude that z < 1
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z < 2
There are several values of w and z that satisfy statement 2 (and the given info). Here are two:
Case a: w = 1 and z = 0. Notice that this satisfies the given info that wz < 2. In this case z < 1
Case b: w = 1 and z = 1.5. Notice that this satisfies the given info that wz < 2. In this case z > 1
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

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Joined: 18 Jan 2017
Posts: 36
Re: If wz < 2, is z < 1? [#permalink]

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10 Apr 2017, 23:41
It is given in (1) that w > 2

We are given that wz < 2

Since we know from (1) that w is positive, we can divide both sides of the inequality by w

=> z < 2/w

Now, since w > 2, this means that 2/w will be lesser than 1
=> z < 1

So, we can conclude that (1) is sufficient.
Re: If wz < 2, is z < 1?   [#permalink] 10 Apr 2017, 23:41
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