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Manager  Joined: 05 Oct 2008
Posts: 216
Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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4

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

n^2 < 1/100
-1/10 < n < 1/10
n is negative so forget right side of equation
multiply both sides by 1/n
1/n * -1/10 < n * 1/n
flip signs as n is negative
1/10n > 1
note 1/n * -1/10 = +1/10n
now multiply both sides by 10

10 * 1/10n < 1 * 10
Don't flip signs
1/n < 10

Originally posted by study on 25 Nov 2011, 02:02.
Last edited by study on 25 Nov 2011, 22:59, edited 3 times in total.
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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study wrote:

n^2 < 1/100
-1/10 < n < 1/10
n is negative so forget right side of equation
multiply both sides by 1/n
1/n * -1/10 < n * 1/n
-1/10n < 1
now multiply both sides by -10
-10 * -1/10n < 1 * -10
Flip signs
1/n > -10

First, that is not what E says (E says that 1/n is greater than positive 10, not negative 10), so that might have suggested that your answer wasn't quite right. I've highlighted your mistake in red. When you multiply on both sides of the inequality by 1/n, you must reverse the inequality, because 1/n is negative.
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Manager  Joined: 05 Oct 2008
Posts: 216
Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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Thanks, Ian. I edited the post. That was a typo. Would you please look at the post now and correct my mistake. Thanks.
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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study wrote:
Thanks, Ian. I edited the post. That was a typo. Would you please look at the post now and correct my mistake. Thanks.

In your edited post, the negative sign vanished - it still needs to be there. You have:

-1/10 < n

Now if we multiply by 1/n on both sides, we must reverse the inequality, since 1/n is negative:

-1/10n > 1

Now we can multiply by 10 on both sides:

-1/n > 10

Now we can multiply by -1 on both sides, reversing the inequality since we are multiplying by a negative:

1/n < -10

Note that you can also do this problem very quickly by finding any suitable number for n (say -1/100) and working out the reciprocal of n.
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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study wrote:

Yes, the negative sign vanished, cuz negative * negative = positive. So a negative 1/n * negative 1/10 = + 1/10n
And that is exactly the part I don't understand. Why would you retain a negative after multiplying a negative number by another negative number? Would you please explain
To illustrate
-2 * -5 = 10. Not -10
so why would -1/n * -1/10 = -1/10n?

If you multiply, say, -1 by x, the result is -x. It makes no difference if x is positive or negative. If x is negative, then -x is a *positive* number, even if it might look negative because of the negative sign in front. There is a second negative sign 'hidden' inside of 'x'.

That's the issue with the step you took in your edited post. When you multiply 1/n by -1/10, the result is *always* equal to -1/10n. It makes no difference at all if n is positive or negative. If n is negative, then -1/10n is a positive number, because you have two negatives in the fraction, one in the numerator and one in the denominator (since n is negative).

I'd strongly suggest you review this part of algebra, because it's fundamental in many GMAT questions.
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Intern  Joined: 09 Feb 2012
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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can you explain why are you flipping sign?

n> - 1/10.

why 1/n<-10 ?

if n=4 for example 4>-1/10 and 1/4>-10. why to flip sign?
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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akakhidze wrote:
can you explain why are you flipping sign?

n> - 1/10.

why 1/n<-10 ?

if n=4 for example 4>-1/10 and 1/4>-10. why to flip sign?

Since n denotes a number to the left of 0 on the number then n is negative, so it can not be 4 as you assumed.

Next, when you multiply (or divide) an inequality by a negative number you must flip the sign of the inequality.

So, for $$n>-\frac{1}{10}$$ --> multiply by negative -10 and flip the sign: $$-10n<1$$ --> divide by negative $$n$$ and flip the sign again: $$-10>\frac{1}{n}$$.

For a complete solution refer to the posts above, for example: if-n-denotes-a-number-to-the-left-of-0-on-the-number-line-91659.html#p667838

Hope it helps.
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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Hi Bunuel
Can it be right if go like the following way?

given (-n)^2<1/100
=>n^2<1/00
=>n<1/10

Now Q ask us to find reciprocal of n
So take the reciprocal on both side simply we get
1/n<10 Ans
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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prasannajeet wrote:
Hi Bunuel
Can it be right if go like the following way?

given (-n)^2<1/100
=>n^2<1/00
=>n<1/10

Now Q ask us to find reciprocal of n
So take the reciprocal on both side simply we get
1/n<10 Ans

No.
Given: n < 0
$$n^2 < (1/100)$$
Note that you will not use -n because n is negative. n already includes the negative sign.
When you take square root, you get |n| < 1/10 (and not n < 1/10). This means -1/10 < n < 1/10. But since n < 0, -1/10 < n < 0.

We need to find the value of reciprocal i.e. 1/n.

n > -1/10
1/n < -10 Note that in an inequality, if both sides of the inequality have the same sign (positive or negative), the sign of inequality (<, >) flips when you take the reciprocal. Here both sides are negative so sign flips.
(Or do what Bunuel did: multiply by -10/n.)
So reciprocal is less than -10.

Instead, I would do this question by thinking of some numbers and figuring out the logic - discussed here: if-n-denotes-a-number-to-the-left-of-0-on-the-number-line-91659.html#p811517
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Re: If n denotes a number to the left of 0 on the number line such that  [#permalink]

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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 such that  [#permalink]

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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 such that  [#permalink]

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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 such that  [#permalink]

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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 such that  [#permalink]

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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 such that  [#permalink]

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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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