What is the value of |x| ? (1) x = - |x| (2) x^2 = 4

SOLUTION

What is the value of |x| ?

(1) x = - |x| --> \(|x|=-x\). This equation holds true for any \(x\) which is less than or equal to zero, so all we know from this statement is that \(x\leq{0}\). Not sufficient.

(2) x^2 = 4 --> \(|x|=2\). Sufficient.

Answer: B.
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Sure. If we do the way you are proposing then we should consider two cases:

If \(x\leq{0}\), then we would have that \(x=-(-x)\) --> \(x=x\), which is obviously true. So, \(x=-|x|\) holds true for any \(x\) which is \(\leq{0}\);


If \(x>{0}\), then we would have that \(x=-x\) --> \(2x=0\) --> \(x=0\), which is not a valid solution since we are considering the range when \(x>{0}\). So, if \(x>{0}\) then \(x=-|x|\) has no valid solutions.

Therefore, from the above we have that \(x=-|x|\) holds true only when \(x\leq{0}\).

Now, you could conclude that right away, since we can rewrite \(x=-|x|\) as \(|x|=-x\), which according to the properties of absolute value is true for \(x\leq{0}\).

Hope it's clear.
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Hi,

Difficulty level: 600

Using (1),
x = -|x|
or \(x \leq 0\), Insufficient.

Using (2),
\(x^2 = 4\)
or |x| = 2. Sufficient.

Answer (B)

Regards,

Please confirm if this is right logic to prove x<=0?

from condition 1: x=-|x|

thus, if x>0 ==> x=-x ==> 2x=0 ==> x=0
and if x<0 ==> x=-(-x) ==> x=x...however x<0, then for this condition x will be always less than zero to satisfy x=x

Thus in combination x<=0.

Cyberjadugar/Bunuel,
is Pavan puneets approach right to confirm stmt ( 1) leads us to x<=0 ?
can you just break stmt 1 down for us a little please ?

From (1) we can conclude that \(x\leq{0}\) (check this: what-is-the-value-of-x-136195.html#p1108170), though the approach you are referring to is not precise enough.
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Hi Bunuel, Can you please let me know where I have made the mistake to derive x<=0. It will be really helpful. Thanks.

I understand the answer explanation, so thanks to everyone who contributed. But what I'm confused is when they use the term "value." Does that always mean they are looking for a single number?

When a DS question asks about the value of some variable, then the statement(s) is sufficient ONLY if you can get the single numerical value of this variable.

Hope it's clear.
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First off, this is a "value" DS question therefore in order to be sufficient, we must be able to calculate a specific value for x

S1: x could be a suite of numbers such as: any negative integer or fraction, and 0 --> not sufficient
S2: 2x = 4 - therefore x = 2 --> sufficient because we know the value of x, and can now answer the question stem |x| --> |2| = 2

Let me know if this helps!

Hi,

I've got a big dilemma for this question.

For statement 2
x^2=4

I see 3 solution fitting:
i) x=2 >> 2^2=4
therefore abs(2)=2

ii) x=(-2) >> (-2)^2=4
therefore abs(-2)=2

BUT
iii) sx=qr(4) >>> sqr(4)^2=4. Am I not correct on this? The sterm/question does not say that x has to be an integer right? So sqr(4) can fit in here, isn't it?
therefore abs(sqr(4))=sqr(4)

Therefore statement 2 is insufficient.

Between the OG and Online explanations, I do not understand why sqr(4) could not fit in statement 2.

Thx

\(\sqrt{4}=2\).

\(x^2=4\) means that \(x=\sqrt{4}=2\) or \(x=-\sqrt{4}=-2\). Two solutions.
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STAT 1: x = -|x|
=> |x| = -x
It just tells us that x is a negative number or zero
So, INSUFFICIENT

STAT 2: x^2 = 4
means x = +-2
So, |x| = 2
So, SUFFICIENT

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Absolute Values


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A few students above had difficulty in processing the first statement: x = -|x|

Here's how you can think through this statement visually:

|x| denotes the distance of an unknown number x from the zero point on the number line. Being the distance, |x| is always non-negative. (Please note that it will be wrong to say that the distance |x| is always positive, because the word 'positive' means 'strictly greater than zero'. It is possible that a point lies ON the zero point, thereby making its distance from the zero point equal to zero. )

So, x = (-)(a positive number) = (a negative number)

Or, x can be equal to zero (that is, on the number line, point x lies ON the point zero. Therefore, |x| = distance between 0 and x = 0 as well)

The important takeaway is that when processing equations of the type x = -|x| etc., start by first considering that |x| is non-negative, since it represents the distance of a number from the zero point on the number line.

Hope this helps!

Japinder
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We need to determine the absolute value of x.

Statement One Alone:

x = -|x|


If x = -|x|, then x must be negative or 0. For example, if x = -3, -3 = -|-3|. However, since we do not have an exact value for x, statement one is not sufficient. We can eliminate answer choices A and D.

Statement Two Alone:

x^2 = 4

We can simplify by taking the square root of both sides of the equation:

√x^2 = √4

|x| = 2

Since we have 2 as the value for |x|, this answers the question.

Answer: B
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HI Bunuel

Can you validate my understanding?
(1) x = - |x|
|x| can be a positive or negative no. (based on x>0 or x<0) but since we do not have
an unique value for x, st 1 is insufficient.


(2) x^2 = 4
We took square root on LHS and RHS to get
|x| = 2 (-2 is not possible on GMAT since square root of negative no shall yield complex no and
we deal with only real nos)
Since we got an unique answer, st 2 is sufficient
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|x| is an absolute value of a number, so it CANNOT be negative. |x| can be positive or 0.

When \(x \le 0\) then \(|x|=-x\), or more generally when \(\text{some expression} \le 0\) then \(|\text{some expression}| = -(\text{some expression})\). For example: \(|-5|=5=-(-5)\);

When \(x \ge 0\) then \(|x|=x\), or more generally when \(\text{some expression} \ge 0\) then \(|\text{some expression}| = \text{some expression}\). For example: \(|5|=5\).

(1) says that |x|=-x, thus \(x \le 0\). Not sufficient.
(2) says that x^2 = 4, so x = 2 or x = -2. I any case |x| = 2. Sufficient.

Check complete solution here: https://gmatclub.com/forum/what-is-the- ... l#p1106786

10. Absolute Value

• Theory
  Math: Absolute value (Modulus)
  Absolute Value: Tips and hints
  Properties of Absolute Values on the GMAT
  Absolute modulus : A better understanding
  GMAT Question Patterns: Abolute Value
  The Reason Behind Absolute Value Questions on the GMAT
  
  • Questions
    Inequality and absolute value questions from my collection
    The E-GMAT Question Series on ABSOLUTE VALUE
    DS Questions
    PS Questions

For more check Ultimate GMAT Quantitative Megathread

Hope it helps.
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Hi Bunuel,
if the question had asked, what is the value of X?
then would C be the answer,

thanks

Yes.

From (1) we have that \(x\leq{0}\) and from (2) we have that x = 2 or -2. So, when combining we can get that x = -2.
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