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

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Joined: 02 Aug 2009
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Re: If n denotes a number to the left of 0 on the number line  [#permalink]

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09 Jan 2018, 05:59
dave13 wrote:
AKProdigy87 wrote:

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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09 Jan 2018, 08:09
1
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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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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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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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.

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

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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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24 Nov 2018, 15:05
A video explanation can be found here:

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.)

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

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22 Jul 2019, 03:52

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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03 Aug 2019, 08:04
1
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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05 Aug 2019, 04:18
1
UNSTOPPABLE12 wrote:
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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05 Aug 2019, 06:44
UNSTOPPABLE12 wrote:
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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05 Aug 2019, 23:06
UNSTOPPABLE12 wrote:
UNSTOPPABLE12 wrote:
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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