Last visit was: 23 Jul 2024, 23:13 It is currently 23 Jul 2024, 23:13
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

# Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0

SORT BY:
Tags:
Show Tags
Hide Tags
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 932
Own Kudos [?]: 1558 [2]
Given Kudos: 115
GMAT 1: 790 Q51 V49
GRE 1: Q170 V170
Director
Joined: 24 Oct 2016
Posts: 581
Own Kudos [?]: 1370 [0]
Given Kudos: 143
GMAT 1: 670 Q46 V36
GMAT 2: 690 Q47 V38
GMAT 3: 690 Q48 V37
GMAT 4: 710 Q49 V38 (Online)
Tutor
Joined: 16 Oct 2010
Posts: 15143
Own Kudos [?]: 66821 [3]
Given Kudos: 436
Location: Pune, India
Director
Joined: 21 Feb 2017
Posts: 509
Own Kudos [?]: 1087 [0]
Given Kudos: 1091
Location: India
GMAT 1: 700 Q47 V39
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 $$x=\sqrt{9}=3$$.

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
Attachments

Screenshot 2020-04-02 at 2.45.57 PM.png [ 61.89 KiB | Viewed 9233 times ]

Math Expert
Joined: 02 Sep 2009
Posts: 94589
Own Kudos [?]: 643423 [1]
Given Kudos: 86728
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
1
Bookmarks
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 $$x=\sqrt{9}=3$$.

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.
Director
Joined: 21 Feb 2017
Posts: 509
Own Kudos [?]: 1087 [0]
Given Kudos: 1091
Location: India
GMAT 1: 700 Q47 V39
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Bunuel wrote:
Kritisood wrote:

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?
Math Expert
Joined: 02 Sep 2009
Posts: 94589
Own Kudos [?]: 643423 [0]
Given Kudos: 86728
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Kritisood wrote:
Bunuel wrote:
Kritisood wrote:

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.
VP
Joined: 15 Dec 2016
Posts: 1352
Own Kudos [?]: 221 [0]
Given Kudos: 188
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
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.

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
Tutor
Joined: 16 Oct 2010
Posts: 15143
Own Kudos [?]: 66821 [0]
Given Kudos: 436
Location: Pune, India
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
jabhatta@umail.iu.edu 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.

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.
Intern
Joined: 25 Apr 2020
Posts: 43
Own Kudos [?]: 73 [2]
Given Kudos: 1
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
2
Kudos
[(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

Posted from my mobile device
Director
Joined: 09 Jan 2020
Posts: 953
Own Kudos [?]: 235 [2]
Given Kudos: 432
Location: United States
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
1
Kudos
1
Bookmarks
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.

Tutor
Joined: 10 Jul 2015
Status:Expert GMAT, GRE, and LSAT Tutor / Coach
Affiliations: Harvard University, A.B. with honors in Government, 2002
Posts: 1182
Own Kudos [?]: 2439 [0]
Given Kudos: 274
Location: United States (CO)
Age: 44
GMAT 1: 770 Q47 V48
GMAT 2: 730 Q44 V47
GMAT 3: 750 Q50 V42
GMAT 4: 730 Q48 V42 (Online)
GRE 1: Q168 V169

GRE 2: Q170 V170
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Top Contributor
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.
GMAT Club Legend
Joined: 03 Oct 2013
Affiliations: CrackVerbal
Posts: 4915
Own Kudos [?]: 7816 [0]
Given Kudos: 221
Location: India
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
Top Contributor
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)

Devmitra Sen
GMAT SME
Senior Manager
Joined: 13 Mar 2021
Posts: 333
Own Kudos [?]: 105 [0]
Given Kudos: 226
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|?
Admitted - Which School Forum Moderator
Joined: 25 Oct 2020
Posts: 1124
Own Kudos [?]: 1091 [0]
Given Kudos: 629
Schools: Ross '25 (M\$)
GMAT 1: 740 Q49 V42 (Online)
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?
GMAT Tutor
Joined: 24 Jun 2008
Posts: 4127
Own Kudos [?]: 9463 [1]
Given Kudos: 91
Q51  V47
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]
1
Bookmarks
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.
Manager
Joined: 23 Oct 2020
Posts: 147
Own Kudos [?]: 4 [0]
Given Kudos: 63
GMAT 1: 710 Q49 V38
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0 [#permalink]

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 ?
Tutor
Joined: 16 Oct 2010
Posts: 15143
Own Kudos [?]: 66821 [2]
Given Kudos: 436
Location: Pune, India
Is ((x-3)^2)^(1/2) = 3-x? (1) x 3 (2) -x|x| > 0 [#permalink]
2
Kudos
Namangupta1997 wrote:

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.
Manager
Joined: 23 Oct 2020
Posts: 147
Own Kudos [?]: 4 [0]
Given Kudos: 63
GMAT 1: 710 Q49 V38
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.
Manager
Joined: 12 Aug 2020
Posts: 52
Own Kudos [?]: 6 [0]
Given Kudos: 570
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!
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x 3 (2) -x|x| > 0 [#permalink]
1   2   3
Moderator:
Math Expert
94589 posts