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# If x > 0, what is the least possible value for x + 1/x ?

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Math Expert
Joined: 02 Sep 2009
Posts: 43894
If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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20 Oct 2014, 06:09
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35% (medium)

Question Stats:

59% (00:52) correct 41% (00:56) wrong based on 360 sessions

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Tough and Tricky questions: Algebra.

If x > 0, what is the least possible value for x + 1/x ?

(A) 0.5
(B) 1
(C) 1.5
(D) 2
(E) 2.5
[Reveal] Spoiler: OA

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Joined: 18 Nov 2013
Posts: 77
Location: India
GMAT Date: 12-26-2014
WE: Information Technology (Computer Software)
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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20 Oct 2014, 08:20
I did this question using plugging in.

Any other short way to do this question?
Intern
Joined: 17 Oct 2010
Posts: 1
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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20 Oct 2014, 09:30
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x+1/x =(X^2+1)/x =(X^2+1-2x+2x)/x =(x^2-2x+1)/x +2x/x =(x-1)^2/x +2

As (x-1)^2 is always greater than zero, hence the expression x+1/X will always be greater than or equal to 2. So the least possible value is 2 at x=1
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Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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20 Oct 2014, 21:30
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coolredwine wrote:
I did this question using plugging in.

Any other short way to do this question?

Note that the product of the terms x and 1/x is a constant x*(1/x) = 1
So the sum will be minimum when the terms are equal.

x = 1/x
x^2 = 1
x = 1 (since x must be positive)

Least possible value of x +1/x = 1 + 1/1 = 2
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews SVP Status: The Best Or Nothing Joined: 27 Dec 2012 Posts: 1839 Location: India Concentration: General Management, Technology WE: Information Technology (Computer Software) Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink] ### Show Tags 21 Oct 2014, 02:20 3 This post received KUDOS $$x + \frac{1}{x}$$ x is reciprocal of $$\frac{1}{x}$$ & vice versa. To have $$x + \frac{1}{x}$$ the lowest, we require to find the number whose reciprocal is same as that of it $$x = \frac{1}{x}$$ $$x^2 = 1$$ x = 1 1 is the only number whose reciprocal = 1 Least addition = 1 + 1 = 2 Answer = D _________________ Kindly press "+1 Kudos" to appreciate Non-Human User Joined: 09 Sep 2013 Posts: 13777 Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink] ### Show Tags 23 Nov 2015, 09:42 Hello from the GMAT Club BumpBot! Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos). Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email. _________________ EMPOWERgmat Instructor Status: GMAT Assassin/Co-Founder Affiliations: EMPOWERgmat Joined: 19 Dec 2014 Posts: 11069 Location: United States (CA) GMAT 1: 800 Q51 V49 GRE 1: 340 Q170 V170 Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink] ### Show Tags 25 Nov 2015, 14:24 Hi All, This question is built around the concept of reciprocals. While some Test Takers might be able to spot the logic involved, most would probably need to TEST VALUES so that they can 'see' what the rule/pattern is: We're told that X > 0. We're asked for the LEAST possible value of X + 1/X. To start, X and 1/X are reciprocals, meaning that when you plug in a value for X, you'll get the inverse of that value for 1/X. For example... X = 10 1/X = 1/10 Sum = 10.1 X = 1/2 1/X = 1/(1/2) = 2 Sum = 2.5 From these two example, we can see that the sum of the two terms WILL be greater than 1. It's just a matter of how 'close to 1' we can get the sum... When... X = 1 1/X = 1 So the sum is 1+1 = 2 There's no other example that will get us any lower than 2, so that must be the answer. Final Answer: [Reveal] Spoiler: D GMAT assassins aren't born, they're made, Rich _________________ 760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com # Rich Cohen Co-Founder & GMAT Assassin Special Offer: Save$75 + GMAT Club Tests Free
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Intern
Joined: 27 Oct 2015
Posts: 26
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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12 Dec 2015, 17:25
The trick is to remember that reciprocals of 0<x<1 will be >1
Also we know that x is positive
the minimum value will be at x = 1 and 1/x = 1 i.e. 2
Retired Moderator
Joined: 12 Aug 2015
Posts: 2428
GRE 1: 323 Q169 V154
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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08 Mar 2016, 09:06
Bunuel wrote:

Tough and Tricky questions: Algebra.

If x > 0, what is the least possible value for x + 1/x ?

(A) 0.5
(B) 1
(C) 1.5
(D) 2
(E) 2.5

x+1/x = (x^2 +1) /x = p (say)
= x^2-px+1=0
for the real roots B^2-4AC must be positive
hence p^2-4=9(for least value)
=> p=2 or -2
hene x=2
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Joined: 28 Jan 2017
Posts: 4
Location: India
Concentration: Finance, General Management
WE: Securities Sales and Trading (Mutual Funds and Brokerage)
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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04 Mar 2017, 04:11
Bunuel wrote:

Tough and Tricky questions: Algebra.

If x > 0, what is the least possible value for x + 1/x ?

(A) 0.5
(B) 1
(C) 1.5
(D) 2
(E) 2.5

Aah how I missed Derivatives!

I know it is out of syllabus, but if you take derivative wrt x -> 1 - 1/(x^2) should be 0 for minima (2nd derivative is +ve so minima) -> x = +-1 -> x = 1

Am I the only one who solved this problem using this approach?
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Joined: 16 Oct 2010
Posts: 7955
Location: Pune, India
Re: If x > 0, what is the least possible value for x + 1/x ? [#permalink]

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05 Mar 2017, 21:48
1
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Expert's post
dhavalpgk wrote:
Bunuel wrote:

Tough and Tricky questions: Algebra.

If x > 0, what is the least possible value for x + 1/x ?

(A) 0.5
(B) 1
(C) 1.5
(D) 2
(E) 2.5

Aah how I missed Derivatives!

I know it is out of syllabus, but if you take derivative wrt x -> 1 - 1/(x^2) should be 0 for minima (2nd derivative is +ve so minima) -> x = +-1 -> x = 1

Am I the only one who solved this problem using this approach?

Derivatives are not tested on GMAT so every GMAT question can be solved without Derivatives.
If you remember your Derivatives concepts well, you are free to use them to solve appropriate questions since all you need to do is give the answer, but most people would be ignoring them.
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Re: If x > 0, what is the least possible value for x + 1/x ?   [#permalink] 05 Mar 2017, 21:48
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