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# If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ?

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If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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25 Dec 2012, 23:36
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If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$?

(1) $$u+v >0$$
(2) $$v>0$$
[Reveal] Spoiler: OA

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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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PraPon wrote:
If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$?

(1) $$u+v >0$$
(2) $$v>0$$

Good Question. This is why GMAT is hard. The concepts are very simple but people get lost while trying to apply them. Let's try to think logically here.

$$u(u+v)\neq{0}$$ implies that neither u nor (u + v) is 0.
$$u >0$$

Take statement 2 first since it is simpler:
(2) $$v>0$$

Consider $$\frac{1}{(u+v)} < \frac{1}{u} + v$$
If v is positive, $$\frac{1}{(u+v)}$$ is less than $$\frac{1}{u}$$ whereas $$\frac{1}{u} + v$$ is greater than $$\frac{1}{u}$$.
Hence right hand side is always greater. Sufficient.

(1) $$u+v >0$$

Since u is positive, v can be either 'positive' or 'negative but smaller absolute value than u'.
We have already seen the 'v is positive' case. In that case, right hand side is greater than left hand side.
Let's focus on 'v is negative but smaller absolute value than u'. Say u is 4 and v is -3. Right hand side becomes negative while left hand side is positive. Hence right hand side is smaller.
Not sufficient.

(Edited)
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18138 [15], given: 236 Math Expert Joined: 02 Sep 2009 Posts: 42630 Kudos [?]: 135779 [8], given: 12714 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 26 Dec 2012, 01:42 8 This post received KUDOS Expert's post 7 This post was BOOKMARKED If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? Is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? --> is $$\frac{-v}{u(u+v)} <v$$? (1) $$u+v >0$$. Since $$u+v >0$$ and $$u >0$$, then $$u(u+v)>0$$. Now, if $$v>0$$, then $$\frac{-v}{u(u+v)}<0 <v$$ but if $$v\leq{0}$$, then $$\frac{-v}{u(u+v)}\geq{0}\geq{v}$$. Not sufficient. (2) $$v>0$$. Since $$u >0$$ and $$v>0$$, then $$\frac{-v}{u(u+v)}<0<v$$. Sufficient. Answer: B. _________________ Kudos [?]: 135779 [8], given: 12714 Manager Joined: 13 Oct 2012 Posts: 69 Kudos [?]: -9 [1], given: 0 Concentration: General Management, Leadership Schools: IE '15 (A) GMAT 1: 760 Q49 V46 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 03 Jan 2013, 21:21 1 This post received KUDOS The ques gets reduced to -> uv(u+v) + v > 0? if u and v both are + then the above is true , hence B Kudos [?]: -9 [1], given: 0 Math Expert Joined: 02 Sep 2009 Posts: 42630 Kudos [?]: 135779 [1], given: 12714 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 12 Jul 2016, 02:24 1 This post received KUDOS Expert's post smartguy595 wrote: $$\frac{1}{(u+v)} - \frac{1}{u}=\frac{u-(u+v)}{(u+v)u}=\frac{u-u-v}{(u+v)u}=\frac{-v}{(u+v)u}$$. Hi Bunuel, Can we cross multiply 'u+v' on left side even if we don't know the sign of 'u+v' We are concerned about the sign when dealing with inequalities: we should keep the sign if we multiply by a positive value and flip the sign when we multiply by a negative value. For equations we can multiply by u+v regardless of its sign. Hope it's clear. _________________ Kudos [?]: 135779 [1], given: 12714 Math Expert Joined: 02 Sep 2009 Posts: 42630 Kudos [?]: 135779 [1], given: 12714 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 01 Jul 2017, 01:49 1 This post received KUDOS Expert's post warriorguy wrote: Bunuel wrote: If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? Is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? --> is $$\frac{-v}{u(u+v)} <v$$? (1) $$u+v >0$$. Since $$u+v >0$$ and $$u >0$$, then $$u(u+v)>0$$. Now, if $$v>0$$, then $$\frac{-v}{u(u+v)}<0 <v$$ but if $$v\leq{0}$$, then $$\frac{-v}{u(u+v)}\geq{0}\geq{v}$$. Not sufficient. (2) $$v>0$$. Since $$u >0$$ and $$v>0$$, then $$\frac{-v}{u(u+v)}<0<v$$. Sufficient. Answer: B. When I first looked at this problem, I did not get how to use given condition u(u+v) not equal to 0. I took numbers and build test cases to check. Is that a valid approach in this case or did I just get lucky? 1. $$u(u+v)\neq{0}$$ means that neither u nor u + v is 0. This is given to ensure that the denominators in $$\frac{1}{(u+v)}$$ and $$\frac{1}{u}$$ are not 0 and thus the fractions are defined. 2. For DS questions testing values is good to get insufficiency (one value gives a NO answer, another gives an YES answer, which means that the statement is not sufficient). Getting that a statement is sufficient by testing values is a bit trickier. You can get say YES answer with several values and conclude that the statement is sufficient but there might be some value which is giving a NO answer and you just missed it, thus incorrect conclusion. So, you should test properly and be careful. _________________ Kudos [?]: 135779 [1], given: 12714 Director Joined: 17 Dec 2012 Posts: 623 Kudos [?]: 549 [0], given: 16 Location: India Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 26 Dec 2012, 05:40 PraPon wrote: If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? (1) $$u+v >0$$ (2) $$v>0$$ is $$\frac{1}{(u+v)} < \frac{1}{u} +v$$or is $$(u+v) > \frac{u}{(1+uv)}$$ From (1), (u+v) is positive. So assume v is positive . If v is positive LHS is greater. Assume v=0. If v=0, then both are equal.So not sufficient. We simply see that since $$v>0$$ and$$u>0$$, $$\frac{1}{u} > \frac{1}{(u+v)}$$. So obviously the RHS is greater from (2) alone. Note: with u positive, saying v is positive is more restrictive than saying (u+v) is positive because in the latter V can be positive or negative. So it is an indication that the answer is likely B. _________________ Srinivasan Vaidyaraman Sravna http://www.sravnatestprep.com/regularcourse.php Premium Material Standardized Approaches Kudos [?]: 549 [0], given: 16 Manager Joined: 26 Dec 2011 Posts: 113 Kudos [?]: 37 [0], given: 17 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 01 Jan 2013, 06:41 Hi Bunuel, if I further simplify your simplified question, i.e. -v/u(u+v) = v to.. by cancelling v on both sides... -1/u(u+v) < 1, then cross multiplying u, given that it is positive the sign does not change...and the question becomes is -1/u+v < u.. ? then I get answer from the condition A as well. given that LHS is negative.. it will be always that u, given that u>0.. what am I doing here wrong? can't I cancel v? Kudos [?]: 37 [0], given: 17 Math Expert Joined: 02 Sep 2009 Posts: 42630 Kudos [?]: 135779 [0], given: 12714 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 02 Jan 2013, 03:20 pavanpuneet wrote: Hi Bunuel, if I further simplify your simplified question, i.e. -v/u(u+v) = v to.. by cancelling v on both sides... -1/u(u+v) < 1, then cross multiplying u, given that it is positive the sign does not change...and the question becomes is -1/u+v < u.. ? then I get answer from the condition A as well. given that LHS is negative.. it will be always that u, given that u>0.. what am I doing here wrong? can't I cancel v? Never multiply (or reduce) an inequality by variable (or by an expression with variable) if you don't know its sign. So you cannot divide both parts of inequality -v/u(u+v)<v by v as you don't know the sign of this unknown: if v>0 you should write -1/u(u+v)>1 BUT if v<0 you should write -1/u(u+v) >1. Hope it helps. _________________ Kudos [?]: 135779 [0], given: 12714 Manager Joined: 21 Jul 2012 Posts: 68 Kudos [?]: 8 [0], given: 32 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 25 Mar 2013, 15:43 Bunuel wrote: If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? Is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? --> is $$\frac{-v}{u(u+v)} <v$$? (1) $$u+v >0$$. Since $$u+v >0$$ and $$u >0$$, then $$u(u+v)>0$$. Now, if $$v>0$$, then $$\frac{-v}{u(u+v)}<0 <v$$ but if $$v\leq{0}$$, then $$\frac{-v}{u(u+v)}\geq{0}\geq{v}$$. Not sufficient. (2) $$v>0$$. Since $$u >0$$ and $$v>0$$, then $$\frac{-v}{u(u+v)}<0<v$$. Sufficient. Answer: B. Bunuel, how did you know to subtract 1/u first? For example, why didnt you add 1/u +v on the right hand side first? When I did this it became extremely messy and over two minutes so I had to guess, but I was hoping to be able to avoid this next time and easily see which form it needs to be in. Thanks for your help! Kudos [?]: 8 [0], given: 32 Manager Joined: 21 Jul 2012 Posts: 68 Kudos [?]: 8 [0], given: 32 Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 26 Mar 2013, 04:55 VeritasPrepKarishma wrote: PraPon wrote: If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? (1) $$u+v >0$$ (2) $$v>0$$ Good Question. This is why GMAT is hard. The concepts are very simple but people get lost while trying to apply them. Let's try to think logically here. $$u(u+v)\neq{0}$$ implies that neither u nor (u + v) is 0 hence v is also not 0. $$u >0$$ Take statement 2 first since it is simpler: (2) $$v>0$$ Consider $$\frac{1}{(u+v)} < \frac{1}{u} + v$$ If v is positive, $$\frac{1}{(u+v)}$$ is less than $$\frac{1}{u}$$ whereas $$\frac{1}{u} + v$$ is greater than $$\frac{1}{u}$$. Hence right hand side is always greater. Sufficient. (1) $$u+v >0$$ Since u is positive, v can be either 'positive' or 'negative but smaller absolute value than u'. We have already seen the 'v is positive' case. In that case, right hand side is greater than left hand side. Let's focus on 'v is negative but smaller absolute value than u'. Say u is 4 and v is -3. Right hand side becomes negative while left hand side is positive. Hence right hand side is smaller. Not sufficient. Answer (B) u(u+v)\neq{0} implies that neither u nor (u + v) is 0 hence v is also not 0. I am confused why v cannot be 0? if u=5 and v=0, then 5(5+0) does not equal 0 and v=0 and it seems to meet the conditions? Kudos [?]: 8 [0], given: 32 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7799 Kudos [?]: 18138 [0], given: 236 Location: Pune, India Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 27 Mar 2013, 19:56 jmuduke08 wrote: VeritasPrepKarishma wrote: PraPon wrote: If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? (1) $$u+v >0$$ (2) $$v>0$$ Good Question. This is why GMAT is hard. The concepts are very simple but people get lost while trying to apply them. Let's try to think logically here. $$u(u+v)\neq{0}$$ implies that neither u nor (u + v) is 0 hence v is also not 0. $$u >0$$ Take statement 2 first since it is simpler: (2) $$v>0$$ Consider $$\frac{1}{(u+v)} < \frac{1}{u} + v$$ If v is positive, $$\frac{1}{(u+v)}$$ is less than $$\frac{1}{u}$$ whereas $$\frac{1}{u} + v$$ is greater than $$\frac{1}{u}$$. Hence right hand side is always greater. Sufficient. (1) $$u+v >0$$ Since u is positive, v can be either 'positive' or 'negative but smaller absolute value than u'. We have already seen the 'v is positive' case. In that case, right hand side is greater than left hand side. Let's focus on 'v is negative but smaller absolute value than u'. Say u is 4 and v is -3. Right hand side becomes negative while left hand side is positive. Hence right hand side is smaller. Not sufficient. Answer (B) u(u+v)\neq{0} implies that neither u nor (u + v) is 0 hence v is also not 0. I am confused why v cannot be 0? if u=5 and v=0, then 5(5+0) does not equal 0 and v=0 and it seems to meet the conditions? Actually, there is no reason why v shouldn't be 0. The only thing is that (u + v) should not be 0. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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04 Jun 2014, 23:04
jmuduke08 wrote:
Bunuel wrote:
If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$?

Is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? --> is $$\frac{-v}{u(u+v)} <v$$?

(1) $$u+v >0$$. Since $$u+v >0$$ and $$u >0$$, then $$u(u+v)>0$$. Now, if $$v>0$$, then $$\frac{-v}{u(u+v)}<0 <v$$ but if $$v\leq{0}$$, then $$\frac{-v}{u(u+v)}\geq{0}\geq{v}$$. Not sufficient.

(2) $$v>0$$. Since $$u >0$$ and $$v>0$$, then $$\frac{-v}{u(u+v)}<0<v$$. Sufficient.

Bunuel, how did you know to subtract 1/u first? For example, why didnt you add 1/u +v on the right hand side first? When I did this it became extremely messy and over two minutes so I had to guess, but I was hoping to be able to avoid this next time and easily see which form it needs to be in. Thanks for your help!

I have the same question too. I do not see where the -v is coming from.

1/u+v - 1/u = u - (u+v) / u(u+v) = +v / u(u+v). Where goes something wrong?

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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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05 Jun 2014, 00:40
chrish06 wrote:
jmuduke08 wrote:
Bunuel wrote:
If $$u(u+v)\neq{0}$$ and $$u>0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$?

Is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$? --> is $$\frac{-v}{u(u+v)} <v$$?

(1) $$u+v >0$$. Since $$u+v >0$$ and $$u >0$$, then $$u(u+v)>0$$. Now, if $$v>0$$, then $$\frac{-v}{u(u+v)}<0 <v$$ but if $$v\leq{0}$$, then $$\frac{-v}{u(u+v)}\geq{0}\geq{v}$$. Not sufficient.

(2) $$v>0$$. Since $$u >0$$ and $$v>0$$, then $$\frac{-v}{u(u+v)}<0<v$$. Sufficient.

Bunuel, how did you know to subtract 1/u first? For example, why didnt you add 1/u +v on the right hand side first? When I did this it became extremely messy and over two minutes so I had to guess, but I was hoping to be able to avoid this next time and easily see which form it needs to be in. Thanks for your help!

I have the same question too. I do not see where the -v is coming from.

1/u+v - 1/u = u - (u+v) / u(u+v) = +v / u(u+v). Where goes something wrong?

$$\frac{1}{(u+v)} - \frac{1}{u}=\frac{u-(u+v)}{(u+v)u}=\frac{u-u-v}{(u+v)u}=\frac{-v}{(u+v)u}$$.

Hope it's clear.
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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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05 Jun 2014, 02:29
PraPon wrote:
If $$u(u+v)\neq{0}$$ and $$u >0$$, is $$\frac{1}{(u+v)} < \frac{1}{u} + v$$?

(1) $$u+v >0$$
(2) $$v>0$$

Good question...I have done it quite a few times but get it wrong occasionally because of the long process of simplifying then inequality

The given expression can be written as

$$\frac{1}{u}$$$$+v$$$$-$$ $$\frac{1}{(u+v)} >0$$

or $$\frac{[(u+v)+ u*(v+u) - u]}{u*(u+v)}$$Simplify and we get

$$\frac{v+uv(u+v)}{u(v+u)}$$ >0

Or v*[$$\frac{1}{(u+v)}$$ + 1] > 0

Now St 1 says u+v >0 and we know u>0 but we don't know whether v is greater than zero or not. Note that product of 2 nos is greater than zero if both the nos are of same sign. From St 1 we know that 1 term ie. [$$\frac{1}{(u+v)}$$ + 1] > 0 but we don't know about v and hence not sufficient

St 2 says v > 0 and we know u>0 so u+v>0 and therefore the expression is greater than zero sufficient.

Ans is B
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If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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11 Feb 2015, 17:49

1/(u+v)<1/u+v

1/(u+v) -1/u -v <0

1/(u+v) - (1-uv/u) <0

u-(u+v) - uv( u+v)/(u+v)u<0

u-u-uv-uv/u<0

-2uv/u <0

-2v<0 or does v positive?

can I devided both sides by -2 and it will be does v>o ?
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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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11 Feb 2015, 20:21
23a2012 wrote:

1/(u+v)<1/u+v

1/(u+v) -1/u -v <0

1/(u+v) - (1-uv/u) <0

u-(u+v) - uv( u+v)/(u+v)u<0

u-u-uv-uv/u<0

-2uv/u <0

-2v<0 or does v positive?

can I devided both sides by -2 and it will be does v>o ?

You have messed up the calculations a bit:

1/(u+v) - (1-uv/u) <0

After this step,
1/(u+v) - 1/u + uv/u < 0

[u - (u + v) + uv(u + v)]/u(u + v) < 0

u-(u+v) - uv( u+v)/(u+v)u<0
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18138 [0], given: 236 Manager Status: Kitchener Joined: 03 Oct 2013 Posts: 96 Kudos [?]: 26 [0], given: 144 Location: Canada Concentration: Finance, Finance GPA: 2.9 WE: Education (Education) Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 12 Feb 2015, 05:46 VeritasPrepKarishma wrote: 23a2012 wrote: I answered the above equestion as follow please correct me if I am wrong 1/(u+v)<1/u+v 1/(u+v) -1/u -v <0 1/(u+v) - (1-uv/u) <0 u-(u+v) - uv( u+v)/(u+v)u<0 u-u-uv-uv/u<0 -2uv/u <0 -2v<0 or does v positive? can I devided both sides by -2 and it will be does v>o ? You have messed up the calculations a bit: 1/(u+v) - (1-uv/u) <0 After this step, 1/(u+v) - 1/u + uv/u < 0 [u - (u + v) + uv(u + v)]/u(u + v) < 0 Instead, you have u-(u+v) - uv( u+v)/(u+v)u<0 OK, I see my mistake here so it will be u-(u+v)+uv(u+v)/u(u+v)<o u-u-v+uv(u+v)/u(u+v)<0 -v+uv/u<0 v(-1+u)/u<0 is that correct? _________________ Click +1 Kudos if my post helped Kudos [?]: 26 [0], given: 144 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7799 Kudos [?]: 18138 [0], given: 236 Location: Pune, India Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink] ### Show Tags 12 Feb 2015, 20:27 23a2012 wrote: OK, I see my mistake here so it will be u-(u+v)+uv(u+v)/u(u+v)<o u-u-v+uv(u+v)/u(u+v)<0 -v+uv/u<0 v(-1+u)/u<0 is that correct? Let me show you the entire calculation since I think we messed up. $$\frac{1}{(u+v)} < 1/u + v$$ $$\frac{1}{(u+v)} - \frac{1}{u} - v < 0$$ $$\frac{u - (u + v) - vu(u + v)}{u(u + v)} < 0$$ $$\frac{u - u - v - vu(u + v)}{u(u + v)} < 0$$ $$\frac{-v - vu(u + v)}{u(u + v)} < 0$$ You cannot simply it further except if you want to separate out the terms. $$\frac{-v}{u(u+v)} - v < 0$$ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ? [#permalink]

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16 Dec 2015, 06:03
SravnaTestPrep

SravnaTestPrep wrote:

is $$\frac{1}{(u+v)} < \frac{1}{u} +v$$or

is $$(u+v) > \frac{u}{(1+uv)}$$

Is this legit unless we know the signs of LHS and RHS?

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Re: If u(u+v) <> 0 and u > 0 is 1/(u + v) < 1/u + v ?   [#permalink] 16 Dec 2015, 06:03

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