If \(x\neq{0}\), then \(\frac{\sqrt{x^2}}{x}=\)


A. -1

B. 0

C. 1

D. x

E. \(\frac{|x|}{x}\)

Thank you. Much clear now.
Also,
Since x is Not equal to zero, is there a value of x for which this equation \(\frac{|x|}{x}\) WILL NOT simplify to \(1\)?

You are asked to find the value of \(\frac {\sqrt{x^2}}{x}\)

Now, for the numerator term:

\(\sqrt{x^2} = x\) or \(-x\)

And the denominator is x.

So if you take the positive value of x as the numerator, the answer is 1, and if you take the negative value, the answer is -1. However, the root symbol specified is used only for positive answers.

Hence the only way to account for this is by using the mod sign. Hence the answer is E.

D is wrong, because either way you look at it \(\frac {\sqrt{x^2}}{x}\) can only be + or - 1

Hope this helps!

You are basically asked to find out what \(\frac{\sqrt{x^2}}{x}\) is.

If it had been given as \(\frac{\sqrt{x}}{x}\) then we can say that the numerator is simply x, and neglect the -x value since the square root sign considers only the positive radical.

But judging by the OA, I think the question was given in terms of what I had written in the first statement. In that case, the numerator is either a +x or a -x, depending on the original value of x. But since we don't know whether the original value was a + or a - number, we use the mod sign to indicate that we are taking the absolute value of the number, which is always positive. So your final answer will be \(\frac{|x|}{x}\)

As an example, let's consider one positive and one negative case.

x = 1
\(x^2\)= 1
\(\sqrt{x^2}\) = x = 1
So here, \(\frac{\sqrt{x^2}}{x}\) = 1

This poses no confusion since the original x value was a positive number by itself.

x = -1
\(x^2\) = 1
\(\sqrt{x^2}\) = (-x) = 1 [Note: The radical sign only indicates that the final result has to be a positive number. This doesn't necessarily mean that the answer is always 'x']
So here, we have \(\frac{\sqrt{x^2}}{x}\) = \(\frac{-x}{x}\) = -1

So, to combine both these results into one answer that fits both, we use \(\frac{|x|}{x}\)

Hope this helps.

Square root of (x^2) ,can be +x or -x and this can be written as |x|.Because Modulus of anything should be always positive as we are just takking the magnitude.If x is +ve ,|x| will be x as its +ve.If x is -ve ,|x| will be -x
as -ve of -ve number will be +ve.

Hence teh answer will be |x|/x

Consider KUDOS if it is helpful to u in some way.

if x>0 then sqrt(x^2) = x

if x<0 then sqrt(x^2) = -x


so we generalize the given expression sqrt(x^2)/x = |x|/x

Answer is E.

Hi Bunuel,

One question here. If x is positive then the answer would be 1. But if x is negitive means x<0. so |-x| = |x| or -(-x) = x which would mean the answer is 1.

Another conceptual question:

If we have in a question square root of any variable (e.g. \(\sqrt{x^2}\)) then the answer would modulus x, provided no information is provided in the question stem.
If we have square root of any number (e.g. \(\sqrt{25}\)) then we take only the positive root, viz. 5 (even without any further information), because that's what the GMAT does. Nevertheless, normally it could be both 5 and negative 5.

Please confirm my understanding, because that's how I read this thread and the GMATClub Math Guide.

Thanks for the clarification!

TITCR. Can't explain any better myself!
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