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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Bunuel wrote:
PriyankaGehlawat wrote:
Bunuel, Thanks for the explanation, but i had one question.
You mentioned : "Couple of things:
The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0."

Can you please let me know why? Because in general square root gives two values, one positive and one negative.



\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).


Hi Bunuel and VeritasKarishma

A follow up to the discussion on roots. If all positive roots give only one answer that is a positive answer then why does \(\sqrt{x^2}\) (positive root) become |x| giving two answers (x could be +ve if x>=0 or -ve if x<0)? Should this also not give only one answer that is +ve?

Maybe i am confused as I have been reading too much on this topic but I am really lost in this regard...

Also, the article shared by ccooley (have attached the snapshot of it) for A) the simplification would be =>|x|/x => |x| = x if x>= 0 OR |x| = -x if x<0

Therefore it would give us two answers -1 and 1. But the article says it will only give us one answer ie -1
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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Kritisood wrote:
Bunuel wrote:
PriyankaGehlawat wrote:
Bunuel, Thanks for the explanation, but i had one question.
You mentioned : "Couple of things:
The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0."

Can you please let me know why? Because in general square root gives two values, one positive and one negative.



\(\sqrt{...}\) is the square root sign, a function (called the principal square root function), which cannot give negative result. So, this sign (\(\sqrt{...}\)) always means non-negative square root.


The graph of the function f(x) = √x

Notice that it's defined for non-negative numbers and is producing non-negative results.

TO SUMMARIZE:
When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the non-negative root. That is:

\(\sqrt{9} = 3\), NOT +3 or -3;
\(\sqrt[4]{16} = 2\), NOT +2 or -2;

Notice that in contrast, the equation \(x^2 = 9\) has TWO solutions, +3 and -3. Because \(x^2 = 9\) means that \(x =-\sqrt{9}=-3\) or \(x=\sqrt{9}=3\).


Hi Bunuel and VeritasKarishma

A follow up to the discussion on roots. If all positive roots give only one answer that is a positive answer then why does \(\sqrt{x^2}\) (positive root) become |x| giving two answers (x could be +ve if x>=0 or -ve if x<0)? Should this also not give only one answer that is +ve?

Maybe i am confused as I have been reading too much on this topic but I am really lost in this regard...

Also, the article shared by ccooley (have attached the snapshot of it) for A) the simplification would be =>|x|/x => |x| = x if x>= 0 OR |x| = -x if x<0

Therefore it would give us two answers -1 and 1. But the article says it will only give us one answer ie -1


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

If x > 0, then \(\sqrt{x^2}=|x|=x=positive\).
If x < 0, then \(\sqrt{x^2}=|x|=-x=-negative=positive\).

As you can see in any case the square root gives positive result.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Bunuel wrote:
Kritisood wrote:

Hi Bunuel and VeritasKarishma

A follow up to the discussion on roots. If all positive roots give only one answer that is a positive answer then why does \(\sqrt{x^2}\) (positive root) become |x| giving two answers (x could be +ve if x>=0 or -ve if x<0)? Should this also not give only one answer that is +ve?

Maybe i am confused as I have been reading too much on this topic but I am really lost in this regard...

Also, the article shared by ccooley (have attached the snapshot of it) for A) the simplification would be =>|x|/x => |x| = x if x>= 0 OR |x| = -x if x<0

Therefore it would give us two answers -1 and 1. But the article says it will only give us one answer ie -1


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

If x > 0, then \(\sqrt{x^2}=|x|=x=positive\).
If x < 0, then \(\sqrt{x^2}=|x|=-x=-negative=positive\).

As you can see in any case the square root gives positive result.


Ok, thanks a lot Bunuel I understood now!! one last thing, is my simplification for option A in the attached image correct? will x take two values in A ie 1 and -1?
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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Kritisood wrote:
Bunuel wrote:
Kritisood wrote:

Hi Bunuel and VeritasKarishma

A follow up to the discussion on roots. If all positive roots give only one answer that is a positive answer then why does \(\sqrt{x^2}\) (positive root) become |x| giving two answers (x could be +ve if x>=0 or -ve if x<0)? Should this also not give only one answer that is +ve?

Maybe i am confused as I have been reading too much on this topic but I am really lost in this regard...

Also, the article shared by ccooley (have attached the snapshot of it) for A) the simplification would be =>|x|/x => |x| = x if x>= 0 OR |x| = -x if x<0

Therefore it would give us two answers -1 and 1. But the article says it will only give us one answer ie -1


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

If x > 0, then \(\sqrt{x^2}=|x|=x=positive\).
If x < 0, then \(\sqrt{x^2}=|x|=-x=-negative=positive\).

As you can see in any case the square root gives positive result.


Ok, thanks a lot Bunuel I understood now!! one last thing, is my simplification for option A in the attached image correct? will x take two values in A ie 1 and -1?


If x < 0, then \(\frac{\sqrt{x^2}}{x}=\frac{|x|}{x}=\frac{-x}{x}=-1\). You can plug any negative number to check it.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
VeritasKarishma wrote:
mrcrescentfresh wrote:
I am having a hard time grasping why we cannot simplify the problem as:

((X-3)^2)^1/2 = (3 - X) to (X - 3)^2 = (3 - X)^2?

I know I have worked problems before where we have been able to solve it by squaring both sides, but when done in this scenario the answer is entirely different than the OA.

Please help.


That is because if the question says:
Is 5 = -5?
And you do not know but you square both sides and get 25 = 25
Can you say then that 5 = -5? No!

I understand that you would have successfully used the technique of squaring both sides before but that would be in conditions like these:
Given equation: \(\sqrt{X} = 3\)
Squaring both sides: X = 9
Here you already know that the equation holds so you can square it. It will still hold. This is like saying:
It is given that 5 = 5.
Squaring both sides, 25 = 25 which is true.


This question is similar to the first case. It is asked whether \(\sqrt{((X-3)^2)} = (3 - X)\)?

LHS is positive because \(\sqrt{((X-3)^2)} = |X-3|\)
and by definition of mod, we know that
|X| = X if X is positive or zero and -X if X is negative (or zero).
Since |X-3| = - (X - 3), we can say that X - 3 <= 0 or that X <= 3
So the question is: Is X <= 3?
Stmnt 1 not sufficient.
But stmnt 2 says -X|X| > 0
This means -X|X| is positive.
Since |X| will be positive, X must be negative to get rid of the extra negative sign in front. So this statement tells us that X < 0. Then X must be definitely less than 3. Sufficient.



Hi VeritasKarishma - can you confirm per the yellow highlight in the above post,

can the question stem be re-phrased as

X <= 3 or x < 3 ?

Per my understanding, the question can be re-phrased as x < 3 only (and not x <= 3)

Reason
|x| = x , if and only if x >=0
and |x| = -x if and only if x < 0

In this question , it is the latter case [|x| = -x if and only if x < 0 ] that shows up

Hence x < 3 should be the re-phrased question and not x <= 3
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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jabhatta@umail.iu.edu wrote:
VeritasKarishma wrote:
mrcrescentfresh wrote:
I am having a hard time grasping why we cannot simplify the problem as:

((X-3)^2)^1/2 = (3 - X) to (X - 3)^2 = (3 - X)^2?

I know I have worked problems before where we have been able to solve it by squaring both sides, but when done in this scenario the answer is entirely different than the OA.

Please help.


That is because if the question says:
Is 5 = -5?
And you do not know but you square both sides and get 25 = 25
Can you say then that 5 = -5? No!

I understand that you would have successfully used the technique of squaring both sides before but that would be in conditions like these:
Given equation: \(\sqrt{X} = 3\)
Squaring both sides: X = 9
Here you already know that the equation holds so you can square it. It will still hold. This is like saying:
It is given that 5 = 5.
Squaring both sides, 25 = 25 which is true.


This question is similar to the first case. It is asked whether \(\sqrt{((X-3)^2)} = (3 - X)\)?

LHS is positive because \(\sqrt{((X-3)^2)} = |X-3|\)
and by definition of mod, we know that
|X| = X if X is positive or zero and -X if X is negative (or zero).
Since |X-3| = - (X - 3), we can say that X - 3 <= 0 or that X <= 3
So the question is: Is X <= 3?
Stmnt 1 not sufficient.
But stmnt 2 says -X|X| > 0
This means -X|X| is positive.
Since |X| will be positive, X must be negative to get rid of the extra negative sign in front. So this statement tells us that X < 0. Then X must be definitely less than 3. Sufficient.



Hi VeritasKarishma - can you confirm per the yellow highlight in the above post,

can the question stem be re-phrased as

X <= 3 or x < 3 ?

Per my understanding, the question can be re-phrased as x < 3 only (and not x <= 3)

Reason
|x| = x , if and only if x >=0
and |x| = -x if and only if x < 0

In this question , it is the latter case [|x| = -x if and only if x < 0 ] that shows up

Hence x < 3 should be the re-phrased question and not x <= 3




|x| = x , if and only if x >=0
and |x| = -x if and only if x < 0
is correct

and so is

|x| = x , if and only if x >0
and |x| = -x if and only if x <= 0

Since |0| = +0 = -0 = 0

You have to take the x = 0 case in one definition, either in |x| = x or in |x| = -x. We usually take the first case but the second is correct too.
This understanding becomes pertinent in a case like this.

When we know that |x| = -x, we know that x is either 0 or negative. Note that x = 0 satisfies here so you cannot ignore it.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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[(x-3)2]1/2 can be x-3 or 3-x..
The first data x not=3 doesn't help as the square root of (x-3)^2 may be x-3 or 3-x...as it depends on the value of x, whether it is smaller than x or greater
The second data says -x|x| is positive. This is possible only if x is negative... because, then -x is positive and |x| is positive and naturally the product is positive. So 3-x will be positive...the square root of the expression had to be positive...and x-3 is negative..so B is sufficient.
Hence B

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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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gmatnub wrote:
Is \(\sqrt{(x-3)^2} = 3-x\)?

(1) \(x\neq{3}\)

(2) \(-x|x| > 0\)

Attachment:
fasdfasdfasdfasdf.JPG


We can re-write the question as: Is \(|x-3| = 3-x\)?

This equation only holds true if \(x ≤ 3\). Think about it. If \(x = 4\), then \(|4-3|\) is not equal to \(3-4\).

Therefore the question we have to answer is: Is \(x ≤ 3\)?

(1) Clearly not sufficient. X could be 4 or x could be 2. INSUFFICIENT.

(2) This statement tells us -x is positive. We can conclude x is negative. SUFFICIENT.

Answer is B.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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A much easier way to approach this non-value DS question is to simply plug in integers and to seek both a Yes and No answer.

Also, even though you can simplify aka "translate" the stem, I don't recommend that you do so, as the original equation works fine, and you run the risk of human error. In the case of this question, it's actually condition #2 that benefits the most from a "translation."

1) Plug in x = 2 to get YES, and x = 4 to get NO. Not sufficient. Cross off A and D.

2) This condition is essentially telling us that x must be negative. Plug in any negative number (-1, -2, -3, etc.) and you will see that it works, and that the answer is always YES. Sufficient.

Hence, the correct answer is Choice B.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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Solution:

We can rephrase or re represent the Question stem here as Is |x−3|=3−x?

This is because sqrt(x^2) =x if x>=0 or -x if x<0

St(1):- x≠3
We have no information on whether it is less than or greater than 3.(Insufficient)

St(2):-−x |x|>0

=> x<0,
=> x<3. Sufficient (option b)

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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Bunuel wrote:
Basically question asks is x ≤ 3?


But why is 0 counted as negative? I have not reflected on this.

|x-3| = |x-3| if x = 3.
|x-3| = |3-x| if x = 3.

If x = 3, then the expressions are both the same. Why do we include |0| as a -|x|?
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
chetan2u Bunuel IanStewart

Initial Approach: Square both sides



\(\sqrt{(x-3)^2} = 3-x\)?

Square both sides

(x-3)\(^{2}\) = (3-x)\(^{2}\)?
x^2 + 9 - 6x = 9 + x^2 - 6x?
0 = 0?
LHS = RHS?

I got stuck on this approach. I know the correct solution now, but why did this simple operation of squaring both side give me such erroneous results?

Is it a rule that you shouldn't square a square root on GMAT or it might screw up your analysis?
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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PyjamaScientist wrote:
Square both sides

I got stuck on this approach. I know the correct solution now, but why did this simple operation of squaring both side give me such erroneous results?

Is it a rule that you shouldn't square a square root on GMAT or it might screw up your analysis?


The fundamental issue here is that the "√" square root symbol does not mean exactly the same thing as the phrase "square root". The number 4, for example, has two square roots, 2 and -2, because if you square either of those numbers, you get 4. But the square root symbol "√" means only the positive square root (or, technically, the non-negative square root if you might be taking the root of zero). Since we must get something positive (or zero) when we apply the "√" to something, we sometimes have to account for that before doing anything else. So if you have this equation:

a = √1

there's obviously no need to do anything to solve, because √1 = 1, so here a = 1 is obviously the only solution. But if instead we decided to solve by squaring both sides, we'd get:

a^2 = 1

and from this new equation, you might think a = 1 and a = -1 are both valid solutions. And they would be if the symbol "√" meant "either the positive and the negative square root", but that's not what it means. It is only equal to one of those two roots. So when we see "a = √(something)", we first need to recognize that a must be at least zero before doing anything else, and then after we square and solve, we need to discard any solutions that are negative. To give a non-trivial example, if you see, say, z^3 = √(z^2), then you might indeed want to square both sides, but before you do that, you have to recognize that z^3, and therefore z, must be 0 or greater, or else you might solve and think that z can equal -1, and it can't. It can only equal 0 or 1 in this case.

Applying that to the GMAT question in this thread, we have this equation:

3 - x = √(x - 3)^2

and since 3 - x is equal to "√(something)", we first need to recognize that 3 - x must be at least zero. So 3 - x > 0, and x < 3. Now you can go ahead and square both sides and solve as you did, and when you arrive at "0 = 0", that just means the equation is always true, for any valid values. Since we know in advance x < 3 must be true, we then can correctly conclude that the equation is true for every value of x that is 3 or less.

That's not how I'd solve this question -- instead we can notice that √(x - 3)^2 = |x - 3| = |3 - x|. So the question is asking: is 3 - x = |3 - x|? If that equation is true, that means the absolute value isn't doing anything (it's not changing "3 - x" into something else; it's leaving it completely unchanged), so this equation is only going to be true if 3 - x is positive or zero, or in other words if x < 3. From that point on, analyzing the statements is straightforward.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
VeritasKarishma Bunuel

How is B sufficient ? It does not cover the case when x = 0,1,2. All these values satisfy the stem equation.

What am I missing ?
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Is ((x-3)^2)^(1/2) = 3-x? (1) x 3 (2) -x|x| > 0 [#permalink]
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Namangupta1997 wrote:
VeritasKarishma Bunuel

How is B sufficient ? It does not cover the case when x = 0,1,2. All these values satisfy the stem equation.

What am I missing ?


Namangupta1997

You are not looking for the exact range of values that will satisfy the inequality given in the question stem. Simplify the question:

Stmnt 2: x is negative

Stmnt 2 tells us that x is negative.

Question stem: Can we say whether x is 3 or greater?
If using statement 2 alone, we can answer the question with a "Yes" or a "No", it will make statement 2 sufficient. If our answer is "May be, may be not", then statement 2 is not sufficient.

Stmnt 2 tells us that x is negative. So we know that for sure it is less than 3. So we can answer the question asked with a "NO, x is not 3 or greater. x is certainly less than 3"
So stmnt 2 is sufficient.

Check these two posts:
https://anaprep.com/algebra-game-must-b ... questions/
https://anaprep.com/algebra-must-be-tru ... questions/

Originally posted by KarishmaB on 05 Dec 2021, 20:50.
Last edited by KarishmaB on 28 Nov 2023, 00:20, edited 1 time in total.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Got it. Thanks VeritasKarishma. It really clears things up. And thanks for the blog links. They are always very insightful. Much appreciated.
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Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x 3 (2) -x|x| > 0 [#permalink]
Can someone post similar questions please? Love this!
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