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(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.

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 wrote:

Hi,

Difficulty level: 600

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

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 ?

pavanpuneet wrote:

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 wrote:

Hi,

Difficulty level: 600

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

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 ?

pavanpuneet wrote:

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 wrote:

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,

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.
_________________

Hi Bunuel, Can you please let me know where I have made the mistake to derive x<=0. It will be really helpful. Thanks.

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 \(x\leq{0}\).

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?

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.

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Stmt 1: no value, stmt2: x = 2 or -2 and !x! = 2, so answer i stmt B alone

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

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.

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.
_________________

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.

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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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
_________________

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

|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.