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

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

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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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Originally posted by EgmatQuantExpert on 19 May 2015, 05:40.
Last edited by EgmatQuantExpert on 07 Aug 2018, 00:30, edited 4 times in total.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 19 May 2015, 10:53
4
1
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 http://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.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post Updated on: 19 May 2015, 20:18
1
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
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Originally posted by cutegirlsimran on 19 May 2015, 08:31.
Last edited by cutegirlsimran on 19 May 2015, 20:18, edited 1 time in total.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 19 May 2015, 10:27
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.

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

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


This is Question 2 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 20th May.

Till then, Happy Solving! :-D

Best Regards
The e-GMAT Team

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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 19 May 2015, 15:35
wakk0 wrote:
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 http://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.




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
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 19 May 2015, 17:57
1
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[/quote]

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
\(\sqrt{(-1^2)} = \sqrt{1} = 1 \neq{x}\) 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.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 19 May 2015, 19:10
2
EgmatQuantExpert wrote:
If x ≠0, then what is the value of \(\frac{(|x|)}{x}\)?

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


This is Question 2 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 20th May.

Till then, Happy Solving! :-D

Best Regards
The e-GMAT Team


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
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post Updated on: 08 Mar 2016, 23:44
2
1
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 --- >\(\sqrt{(-2)^2}\neq(-2)\)
    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
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Originally posted by UJs on 20 May 2015, 01:10.
Last edited by UJs on 08 Mar 2016, 23:44, edited 1 time in total.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post Updated on: 22 May 2015, 18:55
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EgmatQuantExpert wrote:
If x ≠0, then what is the value of \(\frac{(|x|)}{x}\)?

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



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
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Originally posted by D3N0 on 20 May 2015, 01:12.
Last edited by D3N0 on 22 May 2015, 18:55, edited 2 times in total.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 21 May 2015, 13:01
1
EgmatQuantExpert wrote:
If x ≠0, then what is the value of \(\frac{(|x|)}{x}\)?

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


This is Question 2 for the e-GMAT Question Series on Absolute Value.

Provide your solution below. Kudos for participation. The Official Answer and Explanation will be posted on 20th May.

Till then, Happy Solving! :-D

Best Regards
The e-GMAT Team


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.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 21 May 2015, 16:17
1
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
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 22 May 2015, 05:49
2
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.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 22 May 2015, 06:03
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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! :)

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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 22 May 2015, 19:01
EgmatQuantExpert wrote:
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.
Japinder


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.
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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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New post 01 Mar 2016, 19:20
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

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Re: If x ≠0, then what is the value of (|x|)/x ?  [#permalink]

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