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# If 6/(x(x+1))>1, which of the following could the value of x?

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If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Dec 2014, 16:23
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If $$\frac{6}{x(x+1)}>1$$, which of the following could the value of x?

A. -3.5
B. -2.5
C. 2.5
D. 3.5
E. 4.5
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Dec 2014, 19:11
1
Hi viktorija,

This question can be solved by TESTing THE ANSWERS. One (and only one) or those numbers could be a solution to the given inequality, so we could check them (just plug them in) until we find one that "fits" the given inequality.

There is a logical math shortcut here though that we can take advantage of:

We're told that 6/(product) > 1 so the denominator must be LESS than 6. That way 6/(less than 6) will be > 1. So we're really just looking for a product that's less than 6. Logically, we're probably looking for a value for X that's relatively close to 0, so let's check answers B and C....But don't do the math just yet...

Denominator = (-2.5)(-1.5)

Denominator = (2.5)(3.5)

Since the negative signs will cancel out in Answer B, you don't have to do the math to see that Answer B is smaller. Since there's only one answer that will "fit", it has to be B.

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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10 Jan 2015, 02:19
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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10 Jan 2015, 04:26
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?

Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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10 Jan 2015, 06:01
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

$$\frac{6}{x(x+1)} >1$$

let's analyze the denominator. x(x+1) = x^2+x. now x^2+x will always be positive except for numbers lying between -1 and 0. now if x lies between -1 and 0, then the fraction
$$\frac{6}{x(x+1)}$$ will be negative. this violates our initial given condition that $$\frac{6}{x(x+1)} >1$$. hence the expression x^2+x will always be positive.

now since expression x^2+x is positive, therefore we can cross multiply. Thus we have
x^2+x<6
x^2+x-6<0
(x+3)(x-2)<0

now for all values of x which are less than -3, (x+3)(x-2) will always be positive. similarly for all values of x , which are greater than 2, (x+3)(x-2) will always be positive.

hence our desired range is between -3 and 2. i.e. -3<x<2. now out of the given options, only option b lies inside this range. hence answer must be B
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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10 Jan 2015, 06:46
sytabish wrote:
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?

Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!

thanks .
actually we do not need know the sign of X here
Given 6/(x(x+1))>1 implies (x(x+1)) is a +ive quantity so we can simply multilply both sides of inequality by (x(x+1))
6 > (x(x+1))
6> x^2 + x
x^2+x-6 < 0
(x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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10 Jan 2015, 13:37
Hi styabish,

You have to be very careful with your assumptions in the Quant section. This specific Number Property WILL show up on Test Day....

(X)(X+1) = positive

This does NOT mean that X has to be positive.

X COULD be positive....

eg
X = 1
(1)(2) = 2

X COULD be NEGATIVE though...

eg
X = -2
(-2)(-1) = 2

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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12 Jan 2015, 21:56
$$\frac{6}{(x(x+1))} > 1$$

$$\frac{6}{x^2 + x} > 1$$

For x = -3.5

$$\frac{6}{12.25-3.5}$$ >> This would be less than 1

For x = -2.5

$$\frac{6}{6.25-2.5}$$ >> This would be greater than 1

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Aug 2015, 08:27
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Aug 2015, 08:36
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)

Let me ask you: what does x < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

As for the solution please see above.

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Aug 2015, 08:41
2
1
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)

The most straightforward method for this type of question will be to use the values in the options and see which one gives you >1 . Only 1 option must satisfy this requirement.

Additionally, for algebraic solution, look below:

Given : $$\frac{6}{x(x+1)} > 1$$ ----> $$\frac{6}{x(x+1)} - 1 > 0$$ ---> $$\frac{6-x^2-x}{x(x+1)} > 0$$ ----> $$\frac{-6+x^2+x}{x(x+1)} < 0$$

$$\frac{(x+3)(x-2)}{x(x+1)} < 0$$ ----> -3<x<-1 or 0<x<2

Only -2.5 lies in this range.

Hope this helps.
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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Updated on: 27 Aug 2015, 02:19
Bunuel wrote:
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)

Let me ask you: what does x < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

As for the solution please see above.

This link sorted my worries! For anyone who struggled in the last bit after factorising refer this:

Thanks Bunuel! Now I know why did you ask me what didx < -3 (x is less than -3) or x < 2 (x is less than 2) even mean?

Originally posted by aggarwalpooja on 21 Aug 2015, 08:58.
Last edited by aggarwalpooja on 27 Aug 2015, 02:19, edited 1 time in total.
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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21 Aug 2015, 09:15
Engr2012 wrote:
aggarwalpooja wrote:
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

isn't this what you get after factorising
x<-3 or x<2
then how is -2.5 the answer? (since -2.5 is greater than -3)

The most straightforward method for this type of question will be to use the values in the options and see which one gives you >1 . Only 1 option must satisfy this requirement.

Additionally, for algebraic solution, look below:

Given : $$\frac{6}{x(x+1)} > 1$$ ----> $$\frac{6}{x(x+1)} - 1 > 0$$ ---> $$\frac{6-x^2-x}{x(x+1)} > 0$$ ----> $$\frac{-6+x^2+x}{x(x+1)} < 0$$

$$\frac{(x+3)(x-2)}{x(x+1)} < 0$$ ----> -3<x<-1 or 0<x<2

Only -2.5 lies in this range.

Hope this helps.

Thanks Engr12, plugging in the value is guess the easiest!
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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02 Jan 2017, 13:43
1
$$\frac{6}{(x(x+1))} > 1$$

6 > x(x+1)
$$6 > x^{2} + x$$
Plug In options
B
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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20 Feb 2018, 11:10
Lucky2783 wrote:
sytabish wrote:
Lucky2783 wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

Given 6/(x(x+1))>1 i.e x(x+1) is +ive implies X is positive ;
since we are dealing with all positives
6/(x(x+1))>1 ---> (x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

Bunuel am i doing right ?

Lucky2783 ,I don't think this derivation of yours is right: x(x+1) is +ive implies X is positive. Please check!

thanks .
actually we do not need know the sign of X here
Given 6/(x(x+1))>1 implies (x(x+1)) is a +ive quantity so we can simply multilply both sides of inequality by (x(x+1))
6 > (x(x+1))
6> x^2 + x
x^2+x-6 < 0
(x+3)(x-2)<0
++++++(-3)----(2)++++++
so anything between -3 and 2 satisfies the inequality .
B. -2.5

hello there!
How did you manage to draw the line based on this (x+3)(x-2)<0 is there rule to transform it into line ?
thank you
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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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13 Sep 2018, 16:24
1
Hi dave13,

Let me try to respond to your query.

If $$\frac{6}{x(x+1)}$$>1, which of the following could the value of x?

note x and (x+1) are two consecutive numbers, then their product is always positive . Why
if x is negative then (x+1) is also negative and their product is always positive
if x is positive then (x+1) is also positive and their product is always positive
But x cannot be -1 and 0 , because the exp will be undefined for these values.

So
we can write $$\frac{6}{x(x+1)}$$>1 as $$\frac{6}{x(x+1)}$$-1>0

So $$\frac{(6-x(x+1)}{x(x+1)}$$>0

$$\frac{(6-x^2-x)}{x(x+1)}$$>0

$$\frac{-((x+3)(x-2))}{x(x+1)}$$>0

When we multiply by -1 on both sides we change the sign of inequality.

$$\frac{((x+3)(x-2))}{x(x+1)}$$<0

Now if you draw the number line and have positive and negative regions this is how it would look

++++++++++++(-3)---------------------(-1)++++++++++++(0)--------------------(2)++++++++++

Now the region where the inequality holds is
-3<x<-1 and 0<x<2

Now options B, C, D are greater than 2 so discard.
Option A is less than -3 so discard
Option B lies between -3<x<-1 so this could be possible value of for which the inequality will hold.

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Re: If 6/(x(x+1))>1, which of the following could the value of x?  [#permalink]

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14 Sep 2018, 06:09
viktorija wrote:
If 6/(x(x+1))>1, which of the following could the value of x?

A.-3.5
B.-2.5
C.2.5
D.3.5
E.4.5

$$\frac{6}{[x (x + 1)]}$$$$> 1$$

$$\frac{6}{[x^2 + x]}$$ > 1

We know that $$x^2 + x$$ can never be negative irrespective of the value of $$"x"$$, therefore

$$6 > x^2 + x$$

Or $$x^2 + x - 6 < 0$$

$$(x + 3) (x - 2) < 0$$

$$(x - 2) < 0$$ or $$(x + 3) > 0$$

$$x < 2$$ or $$x > -3$$

$$-3 < x < 2$$

Answer : B = $$-2.5$$
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Re: If 6/(x(x+1))>1, which of the following could the value of x?   [#permalink] 14 Sep 2018, 06:09
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