If x ≠0, then what is the value of (|x|)/x ?

Question

Question 2 for The e-GMAT Question Series on Absolute Value.

If x ≠0, then what is the value of \(\frac{(|x|)}{x}\)?

(1) \(\sqrt{(x^2)}=x\)
(2) \(|x-4| = \frac{x}{3}\)

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Official Answer

D

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wakk0 · Accepted Answer

From the question: if x < 0 then \frac{(|x|)}{x} = -1 since you would have a \frac{pos}{neg} if x > 0 then \frac{(|x|)}{x} = 1 since you would have a \frac{pos}{pos} So this question boils down to is x pos or neg? Statement 1) \sqrt{(x^2)}=x This means that x must be a positive value so this answers the question. Sufficient The square root function returns a positive value. If you find yourself screaming or wondering "What about the negative value?!?" ;-)

Look here https://gmatclub.com/forum/math-number-theory-88376.html#p666609 and do a search on the page for "even roots" (without quotes). Statement 2) |x-4| = \frac{x}{3} Pulling off the absolute value signs gives us two possibilities: A) x - 4 = \frac{-x}{3} and B) x-4 = \frac{x}{3} Take a look at each one: A) x-4 = \frac{-x}{3} 3x-12 = -x 4x = 12 x = 3 so x is positive and that answers the question. B) x - 4 = \frac{x}{3} 3x - 12 = x 2x = 12 x = 6 so x is positive and that answers the question. Both A & B give me the same answer to the question (positive one) so 2 is sufficient. Statement 1 and 2 are both sufficient so the answer is D.

cutegirlsimran · Answer

Form statement 1 we get : we know from eq. that \sqrt{x^2} is always +ve so we get positive value of x hence we get answer = 1.

From statement 2 we get : x=6 or x=3 so our final answer will be 1 so statement is sufficient

Hence Answer must be D

shriramvelamuri · Answer

My Vote for Option B.

Option A: X can be + and -

Option B: X will be plus- either 6 or 3. So answer will be 1.

rohitd80 · Answer

Hi,
Thanks for the sharing the gmatclub math-number-theory and the GMAT's rule - "Even roots have only a positive value on the GMAT"

I understand that in the expression ((x)^2)^1/2 the solution will always be positive regardless of whether X is +ve or -ve!
But, in reality X (to be substituted in Statement 1) can be -ve or +ve.......... for eg in the case of x = -1 or +1 when substituted in Statement 1) will always yield a +1!

I agree from the analysis of Statement 2) that x will be either 6 or 3 (both positive)

Thanks

wakk0 · Answer

Hi,
Thanks for the sharing the gmatclub math-number-theory and the GMAT's rule - "Even roots have only a positive value on the GMAT"

I understand that in the expression ((x)^2)^1/2 the solution will always be positive regardless of whether X is +ve or -ve!
But, in reality X (to be substituted in Statement 1) can be -ve or +ve.......... for eg in the case of x = -1 or +1 when substituted in Statement 1) will always yield a +1!

I agree from the analysis of Statement 2) that x will be either 6 or 3 (both positive)

Thanks

Regarding your comments about statement 1. I am not quite sure I am following what you are saying. If you plug in a  x = -1  you would get the following
 \sqrt{(-1^2)} = \sqrt{1} = 1 
eq{x}  because we said above that  x = -1 . So, either the equation is wrong, or we picked a bad number. Since we must take statement 1 to be true, then x can not equal a negative number (i.e. we can't plug in a negative number since that makes the statement not true.) Therefore x can only be a positive number.

chetan2u · Answer

hi ,

(1)  \sqrt{(x^2)}=x 
Since the LHS is always positive, x will be positive and therefore the statement is suff..

(2)  |x-4| = \frac{x}{3} 
on solving, the equation gives us two positive value . irrespective of the values, the equation would always be 1.. suff

ans D

UJs · Answer

Q :   If x ≠0, then what is the value of  \frac{(|x|)}{x} ?

(i)  \sqrt{(x^2)}=x 
  |x| = x    . . . Note :\sqrt{(x^2)} = |x|

\frac{(|x|)}{x} = 1   Sufficient

Alternate way

just to be 100% sure , we can try some numbers (use x= 2, x = -2)
 use x = 2   --- >  \sqrt{(2)^2}=(2) 
 use x = -2  --- > \sqrt{(-2)^2}
eq(-2) 
implies x is positive number , hence

\frac{(|x|)}{x} = 1   Sufficient

(ii)  |x-4| = \frac{x}{3} 
 
Solving both side  (x-4)=\frac{x}{3} --> x = 6  &  (x-4) = \frac{-x}{3} --> x =3 
both cases

\frac{(|x|)}{x} = 1   Sufficient

Ans :  D

D3N0 · Answer

Statement 1) |x|/x = 1 ;
Statement 2) solving the inequality gives us 2 values for x and |x|/x ; again 1 here.

Both statement alone are sufficient to ans: D

reto · Answer

Statement 1 means x is positive since any square root in GMAT will be positive. Thus making the statement  \frac{(|x|)}{x}  positive x / positive x > Sufficient

Statement 2 gives values for x (3 and 6), both positive. The division will result always in 1. Sufficient.

Therefore Answer Choice D.

sudh · Answer

Another link to the problem solved by Bunuel to know for sure why eventh root of x is positive. new-algebra-set-149349-60.html#p1200948

EgmatQuantExpert · Answer

Official Explanation

Correct Answer: D

If x > 0 ,  \frac{(|x|)}{x}=1 
If x < 0,  \frac{(|x|)}{x}= -1

So, we need to find if x is positive or negative.

St.1:
 \sqrt{(x^2 )}=x

The key to analysing this statement correctly is to know that the sqrt symbol literally translates as "the positive square root of"
So, St. 1 tells us that x is positive.
Sufficient.

St. 2:
|x-4| = x/3

Left Hand Side is always positive.
So, Right Hand Side must be positive as well.
So, x is positive.
Sufficient.

EgmatQuantExpert · Answer

dkumar2012 UJs chetan2u rohitd80 wakk0 shriramvelamuri cutegirlsimran Great job done in answering this question correctly! :-D

Just one question, and this is an important question: In Statement 2 analysis, did we need to actually solve for x? Remember, the purpose with which we are going to Statements 1 and 2 is simply to find if x is positive or negative. We can know this in St. 2 without solving for the values of x. Important Takeaway: In DS Questions, one should be careful to not waste time on unnecessary calculations. Hope this helped!

Best Regards Japinder

D3N0 · Answer

Actually No, Mod is always positive as it represents distance between two points, in definition. but for the explanation purpose and for practice it is good to solve to bring confidence in oneself.

mvictor · Answer

1 basically states that |x|=x. so |x|/x=1 always.

2. 2 options:
x-4=x/3
3x=x+12
2x=6
so |x|/x=1

x-4=-x/3
3x=12-x
4x=12
x=3
|x|/x=1

D

stonecold · Answer

Excellent Question team  egmat 
Statement 2 i too calculated the value of x 
In future i wont

New concept learnt ..
Thanks

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If x ≠0, then what is the value of (|x|)/x ?

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