If x 0, what is the value of x?

Official Solution:


If \(x ≠ 0\), what is the value of \(x\)?

(1) The distance between \(x\) and \(-x\) on the number line is equal to the distance between \(x^2\) and \(-x^2\)

The distance between \(a\) and \(b\) on the number line is \(|a - b|\). Thus, the statement above means \(|x - (-x)| = |x^2 - (-x^2)|\)

\(|2x| = |2x^2|\)

\(|2x| = |2x|*|x|\)

Reduce by \(|2x|\) (note here that we can safely do that since \(|2x|\neq 0\). We are told \(x ≠ 0\), thus \(|2x|\neq 0\)):

\(1 = |x|\)

\(x\) can be 1 or -1. Not sufficient.

(2) \(x ? |x| = 0\), where \(?\) represents one of the three operations addition, subtraction, or multiplication

If \(?\) is addition we'd have \(x + |x| = 0\), which is true for all non-positive values of \(x\).

If \(?\) is subtraction we'd have \(x - |x| = 0\), which is true for all non-negative values of \(x\).

If \(?\) is multiplication we'd have \(x*|x| = 0\), which is true when \(x=0\) but since we are told that \(x ≠ 0\), then \(?\) cannot be multiplication.

Not sufficient.

(1)+(2) If \(?\) is subtraction then \(x=1\) (from (1)) but if \(?\) is addition then \(x=-1\) (from (1)). Not sufficient.


Answer: E
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(1) x can take 0 or 1 or -1
in all the cases the distance between x and -x on the number line is equal to the distance between x^2 and -x^2
not sufficient

(2) if x = 0 then ? represents any one of the operations
if x=-1 then ? represents addition
if x=1 then ? represent subtraction
not sufficient

on combining also not sufficient
E
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Stmnt 1) x = 1 or -1 . Insuff
Stmnt 2) if addition, x=-1
if subtraction x=1
Hence E

For the first statement,
distance between x and -x is same as x^2 and -x^2
let say x=1, -1, distance is 2
and 1^2 and -1^2, distance is 2
(GMAT number property, -x^2= -1(x)^2 so -1^2= -1)
insuff.

For the second statement, x? |x| = 0, ? represent + / - / x < does that mean only represent one of them, or all three?
then x must be -ve? x-x= 0
I actually don't really understand the second statement.
someone please explain, thank you

x? |x| = 0
? can represent any of the three (+, -, *) basis the value of x
if u consider x as 1 then ? is -
if u consider x as -1 then ? is +
if u consider x as 0 then ? can take any of -,+,*
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so, again, x can be 1 or x can be -1?
insuff.. ?

Correct and hence E
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Par of GMAT CLUB'S New Year's Quantitative Challenge Set

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Statement 1) x can be 1 or -1. Since, the for x=1, and x=-1, the distance between x and -x on the number line is equal to the distance between x^2 and -x^2. Insufficient.

Statement 2) For addition, x+|X|=0, x=-1 or 0 but x ≠ 0, so x=-1, For subtraction, x-|x|=0, x=1. For multiplication, x*|x|=0, x= 0. SO, x can be 1 or -1. Insufficient.

(1)+(2),

x can be 1 or -1. Insufficient.

Option E is the correct answer.

Given, x ≠ 0
DS Question : what is the value of x?

Statement (1): Since distance between x and -x on the number line is equal to the distance between x^2 and -x^2. Means x and x^2 must have same mod value.
This is possible only if x =1,-1,0. Since , x ≠ 0
So, x= 1,-1
NOT SUFFICIENT

Statement (2): x ? |x| = 0, where ? can be +, -, *
If ? = +
x+|x| = 0
x= -|x|
So, x<0


If ? = -
x-|x| = 0
x= |x|
So, x>0

If ? = *
x*|x| = 0
So, x=0
Since it is given that x ≠ 0.
So ?=* is not possible.

So overall either x<0 or x>0.
NOT SUFFICIENT

Combined statement :
x= -1,1
NOT SUFFICIENT

Answer E

x cannot be equal to 0. It is mentioned in the question stem.

hi, but isn't it given that it represents only one of the operations out of the three so ( i assume here that not both by the language) , how do we conclude statement 2 , it can't be multiplication for sure , but it can EITHER be addition or subtraction not both as what i understand from the lang so if its add x<0 and combined with stat (1) x= -1 but if its sub x>0 and combined with stat 2 x=1 . so considering that it can be one of the three operations with stat (1) , i think we should get a definite answer .

please correct my thought process Bunuel chetan2u
thanks in advance