Author 
Message 
TAGS:

Hide Tags

Math Expert
Joined: 02 Sep 2009
Posts: 59124

Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:00
Question Stats:
60% (02:03) correct 40% (01:50) wrong based on 232 sessions
HideShow timer Statistics
Which of the following is always true? A. If \(\frac{1}{x}\) is greater than \(x\), \(x\) is greater than \(x^2\). B. If \(x\) is greater than \(\frac{1}{x}\), \(2x\) is greater than \(x\). C. If \(x\) is greater than \(2x\), \(\frac{1}{x}\) is greater than \(x\). D. If \(x^2\) is greater than \(x\), \(x^3\) is greater than \(x^2\). E. If \(x\) is greater than \(\frac{1}{x}\), \(x^2\) is greater than \(\frac{1}{x}\).
Official Answer and Stats are available only to registered users. Register/ Login.
_________________



Manager
Joined: 26 Jan 2016
Posts: 180

Which of the following is always true?
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 08:53
Quote: Which of the following is always true?
A. If 1/x is greater than x, x is greater than x^2.
B. If x is greater than 1/x, 2x is greater than x.
C. If x is greater than 2x, 1/x is greater than x.
D. If x^2 is greater than x, x^3 is greater than x^2.
E. If x is greater than 1/x, x^2 is greater than 1/x. A Say x=2 then 1/x=1/2 but 2 is not greater than 4. B Say x=1/2 then 1/x=2 but 2x=1 is not greater than 1/2 C Say x=1 then 2x=2 but 1/x=1 is not greater than 1 D Say x=2 then x^2=4 but x^3=8 is not greater than 4 Hence E
_________________
Your Kudos can boost my morale..!!
I am on a journey. Gradually I'll there..!!
Originally posted by kitipriyanka on 24 Jul 2019, 08:21.
Last edited by kitipriyanka on 24 Jul 2019, 08:53, edited 1 time in total.



GMAT Club Legend
Joined: 18 Aug 2017
Posts: 5282
Location: India
Concentration: Sustainability, Marketing
GPA: 4
WE: Marketing (Energy and Utilities)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:26
Which of the following is always true?
A. If 1x1x is greater than xx, xx is greater than x2x2.
B. If xx is greater than 1x1x, 2x2x is greater than xx.
C. If xx is greater than 2x2x, 1x1x is greater than xx.
D. If x2x2 is greater than xx, x3x3 is greater than x2x2.
E. If xx is greater than 1x1x, x2x2 is greater than 1x1x.
1 not true at x=1/2 2. not true at x=1 3. not true at x=1 4. not true at x=2 5. valid always IMO E



Manager
Joined: 21 Jan 2019
Posts: 100

Which of the following is always true?
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 09:21
Quote: Which of the following is always true?
A. If 1x1x is greater than xx, xx is greater than x2x2.
B. If xx is greater than 1x1x, 2x2x is greater than xx.
C. If xx is greater than 2x2x, 1x1x is greater than xx.
D. If x2x2 is greater than xx, x3x3 is greater than x2x2.
E. If xx is greater than 1x1x, x2x2 is greater than 1x1x. POE: A. If 1x1x is greater than xx, xx is greater than x2x2. if \(\frac{1}{x}\) > x for x = 1/2 this eq will always be true. Hence IncorrectB. If xx is greater than 1x1x, 2x2x is greater than xx. let x = 2 then this will always be true as 4 > 2.
C. If xx is greater than 2x2x, 1x1x is greater than xx. let x= 2 then 1/2 > 2 Hence always true. D. If x2x2 is greater than xx, x3x3 is greater than x2x2. for x =2 then 8 > 4 is not true. Hence D is the answer.E. If xx is greater than 1x1x, x2x2 is greater than 1x1x let x =2 hence 4 > 1/2 Hence this will always be true.Answer is D



Manager
Joined: 30 Aug 2018
Posts: 103
Location: India
Concentration: Finance, Accounting
GPA: 3.36
WE: Consulting (Computer Software)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:28
take cases for positive ,negative ,less than 1 and greater than 1 only E holds true in all C fails when number is 1.



Senior Manager
Joined: 05 Mar 2017
Posts: 261
Location: India
Concentration: General Management, Marketing
GPA: 3.6
WE: Marketing (Hospitality and Tourism)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:31
Which of the following is always true?
This is an easy question, you could solve it by doing multiple cases to disprove the equation. I solve it by plugging in numbers like 2, 1/2, 2, 1/2 Hence was able to disprove all but E.
A. If 1x1x is greater than xx, xx is greater than x2x2. This is an incorrect choice.
B. If xx is greater than 1x1x, 2x2x is greater than xx. This is an incorrect choice.
C. If xx is greater than 2x2x, 1x1x is greater than xx. This is an incorrect choice.
D. If x2x2 is greater than xx, x3x3 is greater than x2x2. This is an incorrect choice.
E. If xx is greater than 1x1x, x2x2 is greater than 1x1x. Yes, this is the correct choice.
The answer choice is E.



Director
Joined: 24 Nov 2016
Posts: 783
Location: United States

Which of the following is always true?
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 09:09
Quote: Which of the following is always true?
A. If 1/x is greater than x, x is greater than x2. B. If x is greater than 1/x, 2x is greater than x. C. If x is greater than 2x, 1/x is greater than x. D. If x2 is greater than x, x3 is greater than x2. E. If x is greater than 1x, x2 is greater than 1/x. x: 2x: 1/x: xˆ2: xˆ3: [1] 0.5…1…2…0.25…0.12 [2] 0.5…1…2…0.25…0.12 [3] 2…4…0.5…4…8 [4] 2…4…0.5…4…8 (A) 2 cases [1,4] not always true; (B) 2 cases [2,3] not always true; (C) 2 cases [2,4] not always true; (D) 2 cases [2,3,4] not always true; Answer (E).
Originally posted by exc4libur on 24 Jul 2019, 08:35.
Last edited by exc4libur on 24 Jul 2019, 09:09, edited 1 time in total.



Intern
Joined: 25 Aug 2015
Posts: 45

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:36
The Answer is E.
Test different cases to try to prove each false.
A. Try this with a negative number. Say x is 2. Then, 0.5 is therefore greater than 2, however 2 is not greater than (2)^2 which is 4. Hence, false.
B. Which cases x will be greater than 1/x is when x is a positive integer >1, or x is negative integer between 0 and 1. Hence, test out both cases. e.g. when x is positive 2, then x is greater than 1/2, and 2(2) is greater than 2. If x is .5, then x is greater than 1/.5 which is 2, but 2(0.5) which is 1 is LESS than 0.5. Hence false
C. In this case, x will be greater than 2x if x is a negative number. Test out if x is 0.5, then 1/x is 5 which is LESS than x. Hence false
D. In this case, x^2 will be greater than x if x is either positive integer greater than 1 or negative integer less than 1. However, if it is a negative integer less than 1, then x^3 is LESS than x^2 as x^3 will be negative. Hence false
E. Correct. Which cases x will be greater than 1/x is when x is a positive integer >1, or x is negative integer between 0 and 1. Hence, test out both cases. e.g. when x is positive 2, then x^2 is 4 which is greater than 4. If x is 0.1, then x^2 is .01 which is greater than 0.1



VP
Joined: 20 Jul 2017
Posts: 1080
Location: India
Concentration: Entrepreneurship, Marketing
WE: Education (Education)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 08:40
A. If 1/x is greater than x, x is greater than x^2. > Possible Range is x < 1 or 0 < x < 1 > x is not always greater than x^2 (eg: x = 2, x^2 = 4)  NO
B. If x is greater than 1/x, 2x is greater than x. > Possible Range is 1 < x < 0 or x > 1 > 2x is not always greater than x (eg: x = 0.5, 2x = 1)  NO
C. If x is greater than 2x, 1/x is greater than x. > Possible Range is x < 0 > 1/x is not always greater than x (eg: x = 0.5, 1/x = 2)  NO
D. If x^2 is greater than x, x^3 is greater than x^2. > Possible Range is x < 1 or x > 1 > x^3 is not always greater than x^2 (eg: x = 2, x^3 = 8, x^2 = 4)  NO
E. If x is greater than 1/x, x^2 is greater than 1/x. > Possible Range is 1 < x < 0 or x > 1 > x^2 is always greater than 1/x (eg: x = 0.5, (x^2, 1/x) = (0.25, 2); eg: x = 2, (x^2, 1/x) = (4, 0.5))  YES
IMO Option E
Pls Hit Kudos if you like the solution



Manager
Joined: 28 Jan 2019
Posts: 127
Location: Peru

Which of the following is always true?
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 18:57
In this one, is not a bad idea to try values to discard some options, so we can first start trying with negative numbers: 2 and 1/2
A) If 1/x is greater than x, x is greater than x^2. For x = 2 we have that 1/x = 0.5 and x^2=4, so not always true B) If x is greater than 1/x, 2x is greater than x. For x =1/2 we have that 1/x = 2 , 2x= 1, so not always true C) If x is greater than 2x, 1/x is greater than x. For x = 1/2, 2x=1 1/x = 2, so not always true D) If x^2 is greater than x, x3 is greater than x^2. Not necessarily, since x could be a negative number, so not always true E) If x is greater than 1/x, x^2 is greater than 1/x. Always true
(E) is our answer
Originally posted by Mizar18 on 24 Jul 2019, 08:48.
Last edited by Mizar18 on 24 Jul 2019, 18:57, edited 1 time in total.



Manager
Joined: 08 Jan 2018
Posts: 129

Which of the following is always true?
[#permalink]
Show Tags
Updated on: 24 Jul 2019, 21:25
Let us check option by option:
A. If \(\frac{1}{x}\) is greater than x –> x can be 0.5, 2 x is greater than \(x^2\) > When x = 0.5 NO
B. If x is greater than \(\frac{1}{x}\)> x can be 0.5, 5 2x is greater than x > When x = 0.5 NO
C. If x is greater than 2x > x can be 1, 0.5 \(\frac{1}{x}\) is greater than x > When x = 0.5 NO
D. If \(x^2\) is greater than x > x can be 1, 2 \(x^3\) is greater than \(x^2\) > When x = 1 NO
E. if x is greater than \(\frac{1}{x}\) > x can be 2, 0.5 \(x^2\) is greater than \(\frac{1}{x}\) > Always True
Answer E
Originally posted by Sayon on 24 Jul 2019, 09:00.
Last edited by Sayon on 24 Jul 2019, 21:25, edited 1 time in total.



Manager
Joined: 28 Feb 2014
Posts: 180
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE: Engineering (Education)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:02
For these type of questions we can divide number into 4 zones ie. infinity to 1, 1 to 0, 0 to 1, 1 to infinity
The question is must be true or always true, for each zone the condition has to satisfy
A. If 1/x is greater than x, means x lies between infinity to 1 and 0 to 1, then x is not greater than x^2 on one region.
B. not true when x is between 1 to 0
C. not true when x is between 1 to 0
D. not true when x is between 1 to 0
E is correct.



Manager
Joined: 30 May 2018
Posts: 157
Location: Canada
GPA: 3.8

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:13
Honestly none of the options work, but I chose A because it looked better.
a) 1/x > x, so 1<x<1. Then, x > x^2. This will work for 0<x<1, but not for 1<x<0. So "not always" true, but found it closest.
b) x>1/x, so x>1 & x<1. Then, 2x>x so x>0. Again, does not "always" work
c) x>2x, so x<0. Then, 1/x > x, we get 1<x<1. Does not always work.
d) x^2 > x, so x<0 & x>1. Then, x^3>x^2, we get x>1. Does not work for x<0
e) x>1/x, so x>1 & x<1. Then, x^2>1/x, we get x>1. Does not work for x<1



Manager
Joined: 10 Mar 2019
Posts: 75
Location: Russian Federation
GPA: 3.95

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:21
A. If \(1/x\) is greater than \(x\), \(x\) is greater than \(x^2\).
\(1/x>x\) case 1. \(x\) is positive \(1>x^2\) \(x^21<0\) \((x1)(x+1)<0\) since x is positive thus \(0<x<1\) case 2. x is negative \(1>x^2\) \(x^2>1\) since \(x\) is negative thus\(inf<x<1\)
Clearly, in the second case \(x\) is not greater than \(x^2\).
B. If \(x\) is greater than \(1/x\), \(2x\) is greater than\(x\).
case 1. \(x\) is positive. \(x^2>1\) \((x1)(x+1)>0\) since is positive thus \(1<x<+inf\) case 2. \(x\)is negative. \(x^2>1\) \(x^2<1\) since \(x\) is negative thus \(1<x<0\)
Clearly, in the second case \(2x\) is not greater than \(x\).
C. If \(x\) is greater than \(2x\), \(1/x\) is greater than \(x\).
\(x>2x\) \(x<0\)
Here, if we take \(1<x<0\) then \(1/x\) is not greater than \(x\)
D. If \(x^2\) is greater than \(x\), \(x^3\) is greater than \(x^2\).
\(x^2>x\) \(x*(x1)>0\) \(inf<x<0\); \(1<x<+inf\)
Thus \(x\) can be any negative number, but \(x^3\) is not greater than \(x^2\)
E. If \(x\) is greater than \(1/x\), \(x^2\) is greater than\(1/x\)
case 1. \(x\) is positive. \(x^2>1\) \((x1)(x+1)>0\) since is positive thus \(1<x<+inf\) case 2. \(x\)is negative. \(x^2>1\) \(x^2<1\) since \(x\) is negative thus \(1<x<0\)
Here, both cases are always true.
The answer is E



Senior Manager
Joined: 12 Dec 2015
Posts: 438

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:26
Which of the following is always true?
A. If 1/x is greater than x, x is greater than x^2. > not always true case1: true: 0<x<1, take x = 0.5, 1/0.5>0.5 => 0.5 > 0.5^2 case2: false: x<1, take x = 2, 1/2>2, but 2 < (2)^2
B. If x is greater than 1/x, 2x is greater than x.> not always true case1: true: x>1, take x = 2, 2>1/2 => 2*2 > 2 case2: false: 1<x<0, take x = 0.5, 0.5>1/(0.5), but 2*(0.5) < 0.5
C. If x is greater than 2x, 1/x is greater than x.> not always true case1: true: x <1, take x = 2, 2>2*(2) => 1/(2) > 2 case2: false: 1<x<0, take x = 0.5, 0.5>2*(0.5), but 1/(0.5) < 0.5
D. If x^2 is greater than x, x^3 is greater than x^2.> not always true case1: true: x >1, take x = 2, 2^2>2 => 2^3 > 2^2 case2: false: x<0, take x = 1, (1)^1>1, but (1)^3 < (1)^2
E. If x is greater than 1/x, x^2 is greater than 1/x.> correct: always true case1: true: x>1, take x = 2, 2>1/2 => 2^2 > 1/2 case2: true: 1<x<0, take x = 0.5, 0.5>1/(0.5), but (0.5)^2 > 1/( 0.5) case1: true: if x>0, x >1/x=>x^2>1=> x>1, so x^2>1/x case2: true: if x<0, x >1/x=>x^2<1=>1<x<0, so x <0 & x^2>0 => x^2>1/x



Senior Manager
Joined: 16 Jan 2019
Posts: 498
Location: India
Concentration: General Management
WE: Sales (Other)

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:33
Lets try to eliminate unfavorable cases here
A. If \(\frac{1}{x}\) is greater than \(x\), \(x\) is greater than \(x^2\). Consider \(x=2\), \(x=2, x^2=4\) So \(x<x^2\)
Eliminate (A)
B. If \(x\) is greater than \(\frac{1}{x}\), \(2x\) is greater than \(x\). Consider \(x=\frac{1}{2}\) \(2x=1, x=0.5\) So \(2x<x\)
Eliminate (B)
C. If \(x\) is greater than \(2x\), \(\frac{1}{x}\) is greater than \(x\). Consider \(x=1\) \(\frac{1}{x}=1, x=1\) So, \(\frac{1}{x}=x\)
Eliminate (C)
D. If \(x^2\) is greater than \(x\), \(x^3\) is greater than \(x^2\). Consider \(x=\frac{1}{2}\) \(x^3=\frac{1}{8}, x^2=\frac{1}{4}\) So \(x^3<x^2\)
Eliminate (D)
E. If \(x\) is greater than \(\frac{1}{x}\), \(x^2\) is greater than \(\frac{1}{x}\).
We are left with (E)
Answer is (E)



Senior Manager
Joined: 10 Jan 2017
Posts: 329
Location: India

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:53
IMO correct answer is E  Explanation attached
Attachments
IMG_20190724_220317.JPG [ 708.52 KiB  Viewed 1547 times ]
_________________
Good, better, best. Never let it rest. 'Till your good is better and your better is best. Please hit +1 Kudos if you like my Post.



Intern
Joined: 28 Feb 2018
Posts: 17

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 09:55
Option A: Put x=0.5 , A out Option B: Put x=2 , B out Option C: Put x=0.5, C out Option D: Put x=1, D out
Option E satisfies all the condition. Hence answer is E.



Intern
Joined: 13 Mar 2019
Posts: 28
Location: India

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 10:01
tricky question, we can solve by cases: 4 cases 2,1/2,2,1/2 A. it fails when we take x is 2 B. it fails when we take x is 1/2 C. it fails when we take x is 1/2 D. it fails when we take x is 2
E will always pass for all cases hence ans is E



Manager
Joined: 26 Mar 2019
Posts: 100
Concentration: Finance, Strategy

Re: Which of the following is always true?
[#permalink]
Show Tags
24 Jul 2019, 10:18
Which of the following is always true?
A. If \(1/x\) is greater than \(x\), \(x\) is greater than \(x^2\). \(1/x > x\) when either \(x\) is negative and \(x<1\) or \(x\) is positive and \(x<1\). If \(x=1/3\), then \(1/x=1/(1/3)=3 > x=1/3\) => \(x=1/3 > x^2=(1/3)^2=1/9\) OK If \(x=5\), then \(1/x=1/5 > x=5\) => \(x=5 > x^2=(5)^2=25\) 5 is not greater than 25 Not Correct
B. If \(x\) is greater than \(1/x\), \(2x\) is greater than \(x\). \(x>1/x\) when either \(x\) is positive and \(x>1\) or \(x\) is negative and \(x>1\) If \(x=2\), then \(x=2 > 1/x = 1/2\) => \(2x=2*2=4 > x=2\) OK If \(x = 1/2\), then \(x=1/2 > 1/x=1/(1/2)=2\) => \(2x=2*(1/2)=1 > x=1/2\) 1 is not greater than 1/2 Not Correct
C. If \(x\) is greater than \(2x\), \(1/x\) is greater than \(x\). \(x>2x\) when x is negative If \(x = 1/3\), then \(x=1/3 > 2x=2/3\) => \(1/x=1/(1/3)=3 > 1/3\) 3 is not greater than 1/3 Not Correct
D. If \(x^2\) is greater than \(x\), \(x^3\) is greater than \(x^2\). \(x^2>x\) when \(x\) is negative or when \(x\) is positive and \(x>1\) If \(x=2\), then \(x^2=2^2=4 > x=2\) => \(x^3=2^3=8 > x^2=2^2=4\) OK If \(x=2\), then \(x^2=(2)^2=4 > x=2\) => \(x^3=(2)^3=8 > x^2=(2)^2=4\) 8 is not greater than 4 Incorrect
E. If \(x\) is greater than \(1/x\), \(x^2\) is greater than \(1/x\). As all previous statements were incorrect, this statement is correct. But let us still prove it. If \(x=1/2\), then \(x=1/2 > 1/x=1/(1/2)=2\) => \(x^2=(1/2)^2=1/4 > 1/x=1/(1/2)=2\). If \(x\) is negative, the second statement is always correct as the square of a negative number is a positive number. If \(x\) is positive and less than 1, the condition is not adhered to. If \(x\) is positive and more than 1, then the statement is correct. Correct.
Answer: E




Re: Which of the following is always true?
[#permalink]
24 Jul 2019, 10:18



Go to page
1 2 3 4
Next
[ 70 posts ]



