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If n denotes a number to the left of 0 on the number line

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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 09 Jan 2018, 05:59
dave13 wrote:
AKProdigy87 wrote:
The answer is A.

We are given two pieces of information:

1) \(n^2 -\frac{1}{10}\)

To determine the conditions on the reciprocal of n:

\(\frac{1}{n}

for example if [m]\sqrt{x}\) = 4 then we need to squae both sides of equation

so \(\sqrt{(x)}\) = \((4)^2\)

---> x= 16

now when it comes to the solution above from this equation \(n^2 < \frac{1}{100}\) we get this \(|n| < \frac{1}{10}\) - why ? should not we square both sides as I did in my example ? :?

Why are we applying this formula \(\sqrt{x^2}\) = \(|x|\) if this doesnt look like \(n^2 < \frac{1}{100}\) ? <-- (here we dont have radical sign) why 100 in denominator is reduced by 10 ? shouldn't 100 be multiplied by itself as in my example ? in my example the number 4 turns into 16 and here it get reduced...so I am confused

Also ss there a difference between \(\sqrt{x^2}\) and \(\sqrt{x}\) ?

many thanks ! :)

perhaps you friends can help :? :-) chetan2u , niks18


Always remember the rule -
\(\sqrt{x}=5\) wll have only one solution as x=25
that is \(\sqrt{25}\) will always be 5 because square root cannot be negative

But \(x^2=25\) means x can be 5 or -5
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If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 09 Jan 2018, 08:09
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hi dave13

Quote:
for example if \(\sqrt{x}\) = 4 then we need to squae both sides of equation

so \(\sqrt{(x)}\) = \((4)^2\)

---> x= 16

now when it comes to the solution above from this equation \(n^2 < \frac{1}{100}\) we get this \(|n| < \frac{1}{10}\) - why ? should not we square both sides as I did in my example ? :?


Note: \(\sqrt{n^2}=|n|\), because here \(n\) is a variable and you don't know the value. for eg. \((-2)^2=4\) & \((2)^2=4\)

so if we say \(n^2=4\), then on taking square root of both sides we will have \(n=|2|=2\) or \(-2\)

Hence here for \(n^2 < \frac{1}{100}\) we get \(|n| < \frac{1}{10}\) on taking square root of \(n^2\), because \(n\) is a variable here

Quote:
Why are we applying this formula \(\sqrt{x^2}\) = \(|x|\) if this doesnt look like \(n^2 < \frac{1}{100}\) ? <-- (here we dont have radical sign) why 100 in denominator is reduced by 10 ? shouldn't 100 be multiplied by itself as in my example ? in my example the number 4 turns into 16 and here it get reduced...so I am confused


we are taking square root here and not squaring. as we have \(n^2\) so by taking square root we will get to \(|n|\); \(100=10^2\) and \(\sqrt{10^2}=10\)

Quote:
Also ss there a difference between \(\sqrt{x^2}\) and \(\sqrt{x}\) ?

many thanks ! :)


There is a lot of difference between \(\sqrt{x^2}\) and \(\sqrt{x}\)

let's assume \(x=2\), then \(x^2=4\) and \(\sqrt{4}\) is different from \(\sqrt{2}\)
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 21 Feb 2018, 10:06
Please suggest where I have gone wrong.

n^2 < 1/100

So, 100 < 1/n^2
So, 1/n^2 > 100

So, I am getting C, while right answer is A.
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If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 21 Feb 2018, 10:48
Nived wrote:
Please suggest where I have gone wrong.

n^2 100

So, I am getting C, while right answer is A.


Hi Nived

kindly read the question correctly. the question asks you to find the reciprocal of \(n\) and not of \(n^2\)

what you have got here is the reciprocal of \(n^2\) which is incorrect.

so you have \(\frac{1}{n^2}>100 =>\frac{1}{n^2}-10^2>0\)

\(=>(\frac{1}{n}-10)(\frac{1}{n}+10)>0\)

so you have either \(\frac{1}{n}>10\) or \(\frac{1}{n}<-10\). Now since \(n\) is towards the left of \(0\), i.e. negative so \(\frac{1}{n}>10\) is not possible

Hence \(\frac{1}{n}<-10\)
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 22 Feb 2018, 17:46
topmbaseeker wrote:
If n denotes a number to the left of 0 on the number line such that the square of n is less than 1/100, then the reciprocal of n must be

A. Less than -10
B. Between -1 and -1/10
C. Between -1/10 and 0
D. Between 0 and 1/10
E. Greater than 10


We are given that n denotes a number to the left of 0 on the number line such that the square of n is less than 1/100. This means n is negative, and it is between 0 and -1/10 since (-1/10)^2 = 1/100. Thus, n could be equal to values such as -1/11, -1/12, -1/13, etc. Notice that when we square these values, they are all less than 1/100.

Let’s take the reciprocal of any of these values listed. The reciprocal of -1/11 is -11. Similarly, the reciprocal of -1/12 is -12 and the reciprocal of -1/13 is -13. Since these reciprocals are all less than -10, answer choice A is correct.

Answer: A
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 21 Oct 2018, 20:20
Given n2<100 and n < 0
n2< 100 => 1/n2 > 100 => 1/n2 - 100>0 => (1/n+10)(1/n-10) >0
solution for 1/n is either 1/n less than -10 or greater than 10 . But n is a negative number so 1/n is less than -10
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If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 24 Nov 2018, 15:05
A video explanation can be found here:
https://www.youtube.com/watch?v=lG5iq9xcsws

First note that if n^2 < 1/100, the range of values of n would be

-1/10 < n < 1/10

But that range is further restricted – since we’re told n is to the left of 0, then

-1/10 < n < 0

(Note that C is a trap answer)

Thinking about reciprocals, we move from negative fractions to negative whole numbers. (The correct answer may become easier to visualize if you draw out the number line.)

Answer A
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 22 Jul 2019, 03:52
Sorry, but I am kind of confused about the sign. Please help me with the following.

When we have negative signs on a variable we're trying to find, we flip the equality sign right?
So, first we have this: -1/10 < n
we divide both side by n : -1/10n < 1
then, we multiply both side by 10 : -1/n < 10
why we don't flip the equality sign like this : 1/n > -10 instead of 1/n < -10?
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If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 03 Aug 2019, 08:04
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Gladiator59 chetan2u Bunuel VeritasKarishma
Hello could you provide your assistance regarding my question? It has to do with inequalities.

lets say we have -1<10n (n<0) when we divide by n I know that it should be like this -1/n>10 and then 1/n < -10
But what I want to ask is when we divide by n, which we know is a negative number, why can't we proceed like this, -1<10n (n<0) 1/n>10 since -1/(a negative number) = positive number
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 05 Aug 2019, 04:18
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UNSTOPPABLE12 wrote:
Gladiator59 chetan2u Bunuel VeritasKarishma
Hello could you provide your assistance regarding my question? It has to do with inequalities.

lets say we have -1<10n (n<0) when we divide by n I know that it should be like this -1/n>10 and then 1/n < -10
But what I want to ask is when we divide by n, which we know is a negative number, why can't we proceed like this, -1<10n (n<0) 1/n>10 since -1/(a negative number) = positive number


If we know -1 < 10n
(n is a negative number so the right hand side is negative too. n could take values such as -1/20, -1/50 etc)

To process it easily, we can just divide both sides by 10 (n gets separated out)
-1/10 < n
Since n < 0, we get -1/10 < n < 0

If instead, we divide both sides by n, we get
-1/n > 10
How did you get 1/n > 10?
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 05 Aug 2019, 06:44
VeritasKarishma wrote:
UNSTOPPABLE12 wrote:
Gladiator59 chetan2u Bunuel VeritasKarishma
Hello could you provide your assistance regarding my question? It has to do with inequalities.

lets say we have -1<10n (n<0) when we divide by n I know that it should be like this -1/n>10 and then 1/n < -10
But what I want to ask is when we divide by n, which we know is a negative number, why can't we proceed like this, -1<10n (n<0) 1/n>10 since -1/(a negative number) = positive number


If we know -1 < 10n
(n is a negative number so the right hand side is negative too. n could take values such as -1/20, -1/50 etc)

To process it easily, we can just divide both sides by 10 (n gets separated out)
-1/10 < n
Since n < 0, we get -1/10 < n < 0

If instead, we divide both sides by n, we get
-1/n > 10
How did you get 1/n > 10?


Well, to be frank with you I was really confused between handling inequalities when there is a variable and when we have a specific number what I did was to consider n<0 but then imagined in my head that n is eg.-2 so i both changed the sign of inequality and the sign of the fraction to positive, which is wrong so instead of just dividing -1<10n by n(n<0) and getting -1/n>10 I would divide by n and simultaneously think that n is a number like eg.-2 so I would also change the sign of -1/n to 1/n which is wrong . thank you VeritasKarishma for your reply.
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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New post 05 Aug 2019, 23:06
UNSTOPPABLE12 wrote:
VeritasKarishma wrote:
UNSTOPPABLE12 wrote:
Gladiator59 chetan2u Bunuel VeritasKarishma
Hello could you provide your assistance regarding my question? It has to do with inequalities.

lets say we have -1<10n (n<0) when we divide by n I know that it should be like this -1/n>10 and then 1/n < -10
But what I want to ask is when we divide by n, which we know is a negative number, why can't we proceed like this, -1<10n (n<0) 1/n>10 since -1/(a negative number) = positive number


If we know -1 < 10n
(n is a negative number so the right hand side is negative too. n could take values such as -1/20, -1/50 etc)

To process it easily, we can just divide both sides by 10 (n gets separated out)
-1/10 < n
Since n < 0, we get -1/10 < n < 0

If instead, we divide both sides by n, we get
-1/n > 10
How did you get 1/n > 10?


Well, to be frank with you I was really confused between handling inequalities when there is a variable and when we have a specific number what I did was to consider n<0 but then imagined in my head that n is eg.-2 so i both changed the sign of inequality and the sign of the fraction to positive, which is wrong so instead of just dividing -1<10n by n(n<0) and getting -1/n>10 I would divide by n and simultaneously think that n is a number like eg.-2 so I would also change the sign of -1/n to 1/n which is wrong . thank you VeritasKarishma for your reply.


Yes, you don't have to change the sign of the fraction. Note that -1/n is a positive number because n is negative. When you make it 1/n, you are making it negative again.
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Re: If n denotes a number to the left of 0 on the number line   [#permalink] 05 Aug 2019, 23:06

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