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

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Math Expert
Joined: 02 Aug 2009
Posts: 5518

Kudos [?]: 6409 [0], given: 122

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

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09 Jan 2018, 04:59
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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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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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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, 07:09
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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 $$\sqrt{2}$$

Kudos [?]: 379 [1], given: 42

If n denotes a number to the left of 0 on the number line   [#permalink] 09 Jan 2018, 07:09

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