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# If w > x^2 ≥ y, which of the following must be true?

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Math Expert
Joined: 02 Sep 2009
Posts: 46301
If w > x^2 ≥ y, which of the following must be true? [#permalink]

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21 Mar 2017, 05:42
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Difficulty:

75% (hard)

Question Stats:

46% (01:09) correct 54% (01:17) wrong based on 182 sessions

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If w > x^2 ≥ y, which of the following must be true?

I. x ≥ 0
II. w > |x|
III. w > y ≥ x

A. None
B. II only
C. I and II only
D. I and III only
E. I, II and III

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Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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21 Mar 2017, 06:09
Bunuel wrote:
If w > x^2 ≥ y, which of the following must be true?

I. x ≥ 0
II. w > |x|
III. w > y ≥ x

A. None
B. II only
C. I and II only
D. I and III only
E. I, II and III

w>x^2 => w > |x| …..so II is correct
x can be greater than zero or can be –ve ….so I may or may not be true in all cases
Since x and y are NOT necessarily integers, y>=x may NOT be true in some cases …so III may or may not be true….

Hence Option B (II only)
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If w > x^2 ≥ y, which of the following must be true? [#permalink]

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21 Mar 2017, 11:27
2
mihir0710 wrote:
Bunuel wrote:
If w > x^2 ≥ y, which of the following must be true?

I. x ≥ 0
II. w > |x|
III. w > y ≥ x

A. None
B. II only
C. I and II only
D. I and III only
E. I, II and III

w>x^2 => w > |x| …..so II is correct
x can be greater than zero or can be –ve ….so I may or may not be true in all cases
Since x and y are NOT necessarily integers, y>=x may NOT be true in some cases …so III may or may not be true….

Hence Option B (II only)

w > x^2

what if w=1/3 and x=1/2 or -1/2

|x| > w > x^2

1/2 > 1/3 > 1/4

Hence II. w > |x| is not MUST BE TRUE.

Pattern: whenever a variable is compared with other variable having exponent then:

1. check whether variable with exponents can be a fraction
2. check the whether the exponent is even or odd to decide the sign
3. What about the mod of the variable with even exponent
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Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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15 Apr 2017, 23:12
Bunuel
TIA
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Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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15 Apr 2017, 23:59
ehsan090 wrote:
Bunuel
TIA

Hi ehsan090,

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If w > x^2 ≥ y, which of the following must be true? [#permalink]

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Updated on: 18 Apr 2017, 11:15
1
Correct me if I am wrong here
As per question -
w>x^2>=y
since x^2 cannot be negative but x can be negative for e.g x=-2. Question stem does not reveal any information regarding "y". Hence Statement I might or might not be true
Statement II does not work for fractions. If w=1/3 and x=1/2 then w>x^2 but w<|x|
Statement III does not work for irrational nos. for example if y=2 and x = \sqrt{5}. In this case x^2>=y is satisfied but statement III is not satisfied.

Originally posted by niks18 on 16 Apr 2017, 03:42.
Last edited by niks18 on 18 Apr 2017, 11:15, edited 1 time in total.
Math Expert
Joined: 02 Aug 2009
Posts: 5952
Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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16 Apr 2017, 03:59
1
Bunuel wrote:
If w > x^2 ≥ y, which of the following must be true?

I. x ≥ 0
II. w > |x|
III. w > y ≥ x

A. None
B. II only
C. I and II only
D. I and III only
E. I, II and III

Hi,

If it were a part of exam and time is important
$$w>x^2\geq{y}$$
x ≥ 0 should immediately tell us that x^2 should be POSITIVE but there is no reason why x ≥ 0.
Thus I is surely not MUST be true.
If I is taken out, all choices except A and B can be eliminated.

We have to just check for II now.
w>|x|
Since x can be a fraction between 0and 1.... Here we need not be greater than |x|
example - x is 1/2 and y is 1/3....
y>x^2 but y<x...

If x is -ive or greater than 1..
y will be greater than |x| if y>x^2..

Hence II need not be true always..

ONLY option left is A..

A
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Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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16 Apr 2017, 05:57
HKD1710 wrote:
ehsan090 wrote:
Bunuel
TIA

Hi ehsan090,

HKD1710
Need explanation of option 2 and 3.
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Re: If w > x^2 ≥ y, which of the following must be true? [#permalink]

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02 Aug 2017, 02:43
1
Bunuel wrote:
If w > x^2 ≥ y, which of the following must be true?

I. x ≥ 0
II. w > |x|
III. w > y ≥ x

A. None
B. II only
C. I and II only
D. I and III only
E. I, II and III

w > x^2 ≥ y

So, w must be +ve

I. x can be -ve or +ve , we can't say anything about x
II. w > x^2
So for 1<x<infinity , w >|x|
for 0<x<1, w may or may not be greater than |x|.
III. w>y this may or may be possible
Similarly y>/x this may or may not be possible.

So None of the I, II,III are correct.

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Re: If w > x^2 ≥ y, which of the following must be true?   [#permalink] 02 Aug 2017, 02:43
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