Which of the following COULD be correct?

Question

Which of the following COULD be correct?
 
I.  x < x^3 < x^2  
II.  x < x^3 < x^2 < x^4 
III.  x^3 < x^2 < x^4 
IV.  x^3 < x < x^2 < x^4

A. I only 
B. I and III only
C. I, II, and IV only
D. I, III, and IV only
E. I, II, III, and IV

JonShukhrat · Accepted Answer

Whenever we see such problems we should try four types of numbers:  + integer ,  + fraction ,  -integer , and  -fraction .

If none of them make the inequality true, then it can  NOT  be true. If any of them makes the inequality true, then there is  NO  need to try others, since it already  CAN  be true.

I.   x<x^3<x^2

If  x = \frac{-1}{3} , then  \frac{-1}{3}<\frac{-1}{27}<\frac{1}{9}  is true

II.   x<x^3<x^2<x^4

If  x = 3 , then  3<27<9<81  is  not  true

If  x = \frac{1}{3} , then  \frac{1}{3}<\frac{1}{27}<\frac{1}{9}<\frac{1}{81}  is  not  true

If  x = -3 , then  -3<-27<9<81  is  not  true

If  x = \frac{-1}{3} , then  \frac{-1}{3}<\frac{-1}{27}<\frac{1}{9}<\frac{1}{81}  is  not  true

III.   x^3<x^2<x^4

If  x = -3 , then  -27<9<81  is true

IV.   x^3<x<x^2<x^4

If  x = -3 , then  -27<-3<9<81  is true

Hence  D

Archit3110 · Answer

Which of the following COULD be correct?
 
I. x<x3<x2 
II.x<x3<x2<x4
III. x3<x2<x4
IV. x3<x<x2<x4

A. I only 
B. I and III only
C. I, II, and IV only
D. I, III, and IV only
E. I, II, III, and IV

let x = -1/2 ,
IMO A
 Bunuel  isnt it that in such type of questions we need to test all the given conditions with same value of x? with x=-1/2 only A stands out where as at x=-2 we get option D ? isnt there an anomaly in this question ? question does not specifies that x has to be an integer value ...  GMATinsight  sir please advise ...

abhishekdadarwal2009 · Answer

IMO : D Which of the following COULD be correct? I. xx 1>x>0 and x>1 Sol: 1: it is present in all choices, that means this has to be true, 2:Not possible 3:let x=-2 then x=-2, x^3=-8 x^2=4 and x^4=16 . So -8<4<16. 4:let x=-2, then x=-2, x^3=-8 x^2=4 and x^4=16 . So -8<-2<4<16.

kitipriyanka · Answer

Take x=-2
x^2=4
x^3=-8
x^4=16

This condition satisfies III and IV.
Now if we look at the options with both III and IV only D and E survives.
Lets analyze II and we can conlude
For II to be true, x<x^3<x^2<x^4
but x^3<x^2<x^4 only when x<0 and when x<0 , x^3<x
Therefore, II cannot be true

Hence D

freedom128 · Answer

I. x -2 < -8 < 4 <16 (NO) x=-\frac{1}{2} --> -\frac{1}{2} < -\frac{1}{8} < \frac{1}{4} < \frac{1}{16} (NO) x=\frac{1}{2} --> \frac{1}{2} < \frac{1}{8} < \frac{1}{4} < \frac{1}{16} (NO) x=2 --> 2 < 8 < 4 <16 (NO) This statement must be FALSE for any x. III. x^3 -8 < 4 <16 (YES!) This statement could be TRUE IV. x^3 -8 < -2 < 4 <16 (YES!) This statement could be TRUE Answer is (D) I, III, and IV only

eakabuah · Answer

We just need to establish that statements I, II, III, and IV is valid for a value of x in order to conclude that it could be correct. If the condition does not hold for any possible value, then we can conclude that it cannot be correct.

I. x<x^3<x^2. This is true for -1<x<0. For example x=-1/2. x^2=1/4, x^3=-1/8. 
Since x(-1/2)<x^3(-1/8)<1/4, I could be correct.

II. x<x^3<x^2<x^4
Not true for x>1, not true for x<-1, not true for 0<x<=1, not true for -1<=x<=0. Hence this statement could never be correct.

III. x^3<x^2<x^4
III. is true for x<-1, for example x=-2, x^2=4, x^3=-8, x^4=16. Since x^3(-8)<x^2(4)<x^3(8), III could be correct.

IV. x^3<x<x^2<x^4
This is true for x<-1. For example x=-2, 
x^3(-8)<x(-2)<x^2(4)<x^4(16)
Hence IV. could be correct.

We know that I, III, and IV could be correct. The answer is therefore D.

Posted from my mobile device

unraveled · Answer

Which of the following COULD be correct?

I. x < x3 < x2
II. x < x3 < x2 < x4
III. x3 < x2 < x4
IV. x3 < x < x2 < x4

A. I only
B. I and III only
C. I, II, and IV only
D. I, III, and IV only
E. I, II, III, and IV

Though each option has an equal chance as an answer but evaluating the options here we can say that options A and B have the highest chances of being the answer. ‘E’ deals with all so ignoring it for the time being.

Options ‘C’ and ‘D’ mentions only and have ‘I’ and ‘IV’ common with option ‘E’. Thus, check for the ‘COULD be correct’ condition for ‘II’ and ‘III’ first.

II. x < x3 < x2 < x4
x < x3 when x > 1 OR -1 < x < 0
x3 < x2 when x < 0 OR 0 < x < 1
x2 < x4 when x > 1 OR x < -1

As we have three conditions i.e. when
i) x < -1
ii) -1 < x < 0
iii) 0 < x < 1
iv) x > 1

We have total four conditions. Let’s refer table snapshot:

In totality, x < x3 < x2 < x4 is never true ever. Since ‘II’ is present in options ‘C’ and ’E’ both, option ‘D’ is the answer.

But for completeness check ‘III’:

III. x3 < x2 < x4
Since this is a part of ‘II’ table for ‘II’ suffices to illustrate that this could be true(x < -1).

IMO Answer (D)

PS: Also, ‘I’ can be checked through above table that is true for -1 < x < 0 AND ‘IV’ is true for x < -1 (reverse of x < x3 for x < -1 is true).

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GKomoku · Answer

Which of the following COULD be correct?

I. \(x < x^3 < x^2\)
II. \(x < x^3 < x^2 < x^4\)
III. \(x^3 < x^2 < x^4\)
IV. \(x^3 < x < x^2 < x^4\)

A. I only
B. I and III only
C. I, II, and IV only
D. I, III, and IV only
E. I, II, III, and IV

This question was provided by Experts Global
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GMATinsight · Answer

Archit3110

No, we don't have to test them all for the same value of x cause they may be true for different ranges of x

I. x<x3<x2 this could be true for x = -1/2
II.x<x3<x2<x4 . x<x3 is true for x between 0 and -1 or for values greater than 1 but x<x2<x4 will be true only for x < -1 or x > 1. Since there is no common raneg of values of x therefore it can not be true
III. x3<x2<x4 . This could be true for x = -2
IV. x3<x<x2<x4 . This could be true for x = -2

Answer: Option D

Shahalikhan · Answer

we need to check between option II and III clearly x^2 x>1 or x<-1 now x^3>x^2=> x is negative thus x^3 cannot be less greater than x two is not possible

brt · Answer

If you try with x=-1/2 
only 1 is right 
so ans is 1

Adarsh_24 · Answer

Knowing these graphs helps a lot in these type of problems for me.
Since its symmetric around -1, 0 or 1. Pretty easy to memorize.

After seeing x^3 lower than x^2. Need to concentrate only on -ve side.

m7runner · Answer

→ Since I is present in all options, then would recommend NOT checking if this is true or not (do so in your review though!).  Eliminate A .
→ IV is present in 3 of the remaining 4 options, so let's check if this is possible. For x = -1, IV is true. Thus,  eliminate B .
→ With three options remaining and II and III still to be evaluated, thus we have no option but to evaluate both the statements.
 → For II  x < x^3 < x^2 < x^4 , is not possible for any value of x (try x = 2, -2, 0.2, -0.2). Thus,  eliminate C and E .

Thus option D is the correct choice.

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Which of the following COULD be correct?

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