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# Is x < y ? (1) z^2 < y (2) x < z

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Joined: 02 Sep 2009
Posts: 61302
Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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13 Mar 2017, 07:20
1
10
00:00

Difficulty:

55% (hard)

Question Stats:

57% (01:29) correct 43% (01:31) wrong based on 192 sessions

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Is x < y ?

(1) z^2 < y
(2) x < z

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Joined: 11 Sep 2015
Posts: 4331
Re: Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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13 Mar 2017, 11:28
6
Top Contributor
5
Bunuel wrote:
Is x < y ?

(1) z² < y
(2) x < z

Target question: Is x < y ?

Statement 1: z² < y
No information about x, so statement 1 is NOT SUFFICIENT

Statement 2: x < z
No information about y, so statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
NOTE: the error that many students will make is to assume that z < z², but this is not necessarily the case.
For example, if z = 1/2, then z² = 1/4, in which case z² < z
On the other hand, if z = 3, then z² = 3, in which case z < z²
This feature will create conflicting answers to the target question. Here's what I mean:
There are several values of x, y and z that satisfy BOTH statements. Here are two:
Case a: x = 0.3, y = 0.4, and z = 0.5 (which also means z² = 0.25). Notice that z² < y and x < z, so both conditions are met. In this case x < y
Case b: x = 0.4, y = 0.3, and z = 0.5 (which also means z² = 0.25). Notice that z² < y and x < z, so both conditions are met. In this case x > y
Since we cannot answer the target question with certainty, the combined statements are NOT SUFFICIENT

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Re: Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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15 Mar 2017, 05:30
1
y>z^2
for z>1, we can say that y>z
if 0<z<1 , we cannot say if y>z. take y,z be 0.05, 0.2 and 0.5, 0.2. plus we have no data about x. INSUFFICIENT

St 2: x<z. no idea about y. INSUFFICIENT

St 1 & St 2: as we cannot relate the value of y and z in st 1, using the st2 also we cannot relate x and y. INSUFFICIENT

Option E
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Re: Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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20 Mar 2017, 05:55
1
Bunuel wrote:
Is x < y ?

(1) z^2 < y
(2) x < z

We need to determine whether x < y.

Statement One Alone:

z^2 < y

Since we do not have any information regarding x, statement one is not sufficient to answer the question.

Statement Two Alone:

x < z

Since we do not have any information regarding y, statement two is not sufficient to answer the question.

Statements One and Two Together:

Using the information in statements one and two, we still cannot answer the question. For instance, if y = 10, z = 2, and x = 1, then x is less than y. However, if z = 1/2, x = 1/3, and y = 1/3, then x IS NOT less than y.

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Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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29 Sep 2019, 04:34
Bunuel wrote:
Is x < y ?

(1) z^2 < y
(2) x < z

This is how i solved...

Statement 1 : We don't know the value of x therefore not sufficient.
Statement 2 : We don't know the value of y therefore not sufficient.

Together Statement 1 and 2, ( Statement 1 says z^2 < y ... let z be 0.5 then z^2 = 0.25 ... and y is greater than z^2 let y be 0.26 onwards.... ( SO Z= 0.5 AND Y = 0.26 )
Statement 2 says z is greater than x which means x is less than 0.5 which means X could be 0.4

In this case X > y ( 0.4 > 0.26 ) therfore is Is x < y ? ans is not true.

nOW LETS TEST THIIS with another value ..
But now if we assume the value of z = 2 then z^2 = 2^2 = 4 and y should be greater than 4 lets say 5 ...
so Z= 2 and y = 5

Statement 2 says x < Z so lets say x = 1 in that case x< y or 1 < 5 true ...

Since we cannot answer the question with certainty ans is E
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Is x < y ? (1) z^2 < y (2) x < z  [#permalink]

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29 Sep 2019, 19:22
Bunuel wrote:
Is x < y ?

(1) z^2 < y
(2) x < z

1) and 2) are insufficient alone. We can combine the statements by squaring 2) to get z^2.

However, depending on the sign of x and z we can have $$x^2 < z^2$$ or $$x^2 > z^2$$ so we must entertain both possibilities (at this point a good guess is E because we already have two cases).

In the first case, we have $$x^2 < z^2 < y$$, thus $$x^2 < y$$. This cannot simplify to $$x < y$$ because we can have x = 0.8 and y = 0.7. Thus combining is insufficient.
Knowing that $$x^2 < y$$ cannot simplify to $$x < y$$ can be valuable knowledge for future problems. Since we cannot prove $$x < y$$ with the given information it is insufficient.

The second case occurs happens when x is negative and has a bigger magnitude than z. Hence x must be negative in this case. From (1) we know y is positive, so $$x < y$$.

However, since the first case was already insufficient, the statements combined are insufficient, without needing to work out case 2.

Ans: E
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Is x < y ? (1) z^2 < y (2) x < z   [#permalink] 29 Sep 2019, 19:22
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