GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 21 Sep 2019, 12:35

### 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

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

Author Message
TAGS:

### Hide Tags

Intern
Joined: 16 Apr 2019
Posts: 9
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

04 Jun 2019, 10:05
Dear Brunel,

You said-

"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?

Remember: x2−−√=|x|x2=|x|. Why?

Couple of things:

The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0.

So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?

Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."

My doubt is as follows-

All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.
Manhattan Prep Instructor
Joined: 04 Dec 2015
Posts: 813
GMAT 1: 790 Q51 V49
GRE 1: Q170 V170
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

04 Jun 2019, 10:14
tamalmallick wrote:
Dear Brunel,

You said-

"Is (x−3)2−−−−−−−√=3−x(x−3)2=3−x?

Remember: x2−−√=|x|x2=|x|. Why?

Couple of things:

The point here is that square root function can not give negative result: wich means that some expression−−−−−−−−−−−−−−√≥0some expression≥0.

So x2−−√≥0x2≥0. But what does x2−−√x2 equal to?

Let's consider following examples:
If x=5x=5 --> x2−−√=25−−√=5=x=positivex2=25=5=x=positive;
If x=−5x=−5 --> x2−−√=25−−√=5=−x=positivex2=25=5=−x=positive."

My doubt is as follows-

All we know that sqrt of a number can be positive or negative results both, how you are saying "square root function can not give negative result"? If you kindly answer this question it would be a great help for me. Looking forward to hear you from.

A lot of people ask this question - you're not alone. I'm not Bunuel, but I did write a short article about it once that should clear things up:

https://www.manhattanprep.com/gmat/blog ... -the-gmat/
_________________

Chelsey Cooley | Manhattan Prep | Seattle and Online

My latest GMAT blog posts | Suggestions for blog articles are always welcome!
Manager
Joined: 20 Feb 2018
Posts: 51
Location: India
Schools: ISB '20
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

25 Jun 2019, 00:17
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff

But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0

If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve

=> sqrt ((x-3)^2) = +X-3

=> sqrt ( (x-3) ^2 ) is not equal to 3-x

=> Option B

Yes, the answer for this question is B.

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

Remember: $$\sqrt{x^2}=|x|$$. Why?

Couple of things:

The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$

Back to the original question:

So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?

When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.

When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.

Basically question asks is $$x\leq{3}$$?

(1) $$x\neq{3}$$. Clearly insufficient.

(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.

Hope it helps.

Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?
Math Expert
Joined: 02 Sep 2009
Posts: 58133
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

25 Jun 2019, 00:20
shobhitkh wrote:
Bunuel wrote:
gautamsubrahmanyam wrote:
I understand that 1) is insuff

But for 2) -x|x| > 0 means x cant be +ve => |x| = -x so that -x (-x) = x^2> 0

If x is -ve => (x-3)^2 = X^2+9-6x = (-ve)^2+9-6(-ve) = +ve+9-(-ve) = +ve +9 + (+ve) = +ve

=> sqrt ((x-3)^2) = +X-3

=> sqrt ( (x-3) ^2 ) is not equal to 3-x

=> Option B

Yes, the answer for this question is B.

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

Remember: $$\sqrt{x^2}=|x|$$. Why?

Couple of things:

The point here is that square root function cannot give negative result: wich means that $$\sqrt{some \ expression}\geq{0}$$.

So $$\sqrt{x^2}\geq{0}$$. But what does $$\sqrt{x^2}$$ equal to?

Let's consider following examples:
If $$x=5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=x=positive$$;
If $$x=-5$$ --> $$\sqrt{x^2}=\sqrt{25}=5=-x=positive$$.

So we got that:
$$\sqrt{x^2}=x$$, if $$x\geq{0}$$;
$$\sqrt{x^2}=-x$$, if $$x<0$$.

What function does exactly the same thing? The absolute value function! That is why $$\sqrt{x^2}=|x|$$

Back to the original question:

So $$\sqrt{(x-3)^2}=|x-3|$$ and the question becomes is: $$|x-3|=3-x$$?

When $$x>3$$, then RHS (right hand side) is negative, but LHS (absolute value) is never negative, hence in this case equations doesn't hold true.

When $$x\leq{3}$$, then $$LHS=|x-3|=-x+3=3-x=RHS$$, hence in this case equation holds true.

Basically question asks is $$x\leq{3}$$?

(1) $$x\neq{3}$$. Clearly insufficient.

(2) $$-x|x| >0$$, basically this inequality implies that $$x<0$$, hence $$x<3$$. Sufficient.

Hope it helps.

Hi Bunuel, in the highlighted portion above, how can we deduce that x will be less than 3 if x is less than 0? x can be 1,2 also, right? May be I am missing something. Can you please clarify?

If a number is less than 0, does not it mean that it's less than 3?
_________________
Director
Joined: 24 Oct 2016
Posts: 511
GMAT 1: 670 Q46 V36
GMAT 2: 690 Q47 V38
Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

26 Jun 2019, 04:34
gmatnub wrote:
Is $$\sqrt{(x-3)^2} = 3-x$$?

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

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

Attachment:
fasdfasdfasdfasdf.JPG

Alternative Approach

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

Case 1: |x-3| > 0 => x > 3
x-3 = 3-x?
2x=6?
x=3?
x=3 is not possible ever since x > 3

Case 2: |x-3| <= 0 => x <= 3
-x + 3 = 3-x?
0=0?
LHS = RHS?
This case would always be true since it can't violate any conditions.

Rephrased Q: Is x <= 3?

Stmt 1: x != 3
Doesn't tell anything about x if it's more than or less than 3. Not sufficient.

Stmt 2: -x|x| > 0
That implies x is always negative or x < 0. Hence x < 3 is also true. Sufficient.

Bunuel EducationAisle VeritasKarishma I got this Q wrong with my initial approach (shown below) of squaring both sides. I was wondering whether we can solve this Q by squaring both sides. If not, why not? I'm also confused how x=1 can be transformed with a few steps to give x=1 & -1 (shown below)? I would really appreciate if you could help me improve my understanding on this issue. Thanks!

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?

Not sure how to proceed?

x=1 transforms to x=+1,-1?
x = 1
Square both sides
x^2 = 1
Take square root of both sides
|x| = 1
x = +1, -1
_________________

If you found my post useful, KUDOS are much appreciated. Giving Kudos is a great way to thank and motivate contributors, without costing you anything.
Senior Manager
Joined: 19 Nov 2017
Posts: 254
Location: India
Schools: ISB
GMAT 1: 670 Q49 V32
GPA: 4
Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3  [#permalink]

### Show Tags

30 Aug 2019, 22:20
jan4dday wrote:
Is $$\sqrt{(x-3)^2}=3-x$$?

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

(2) -x|x| > 3

$$\sqrt{(x-3)^2}=3-x$$
This will be true only when x = 3 or x= 2

Statement 1
$$x\neq{3}$$

It might be equal to 2, 4, anything.

Insufficient.

Statement 2
$$-x|x| > 3$$
$$|x|$$ is always +ve
if $$-x|x| > 3$$, then $$-x > 0$$
this means that $$x$$ is -ve
if $$x$$ is -ve, it cannot equal either $$3$$ or $$2$$.

Sufficient.

Hence, B.
_________________

Vaibhav

Sky is the limit. 800 is the limit.

~GMAC
Intern
Joined: 20 Jun 2019
Posts: 41
Re: Is (x-3)^2 =3-x ? (1) x not = 3 (2) -x|x| > 3  [#permalink]

### Show Tags

04 Sep 2019, 21:59
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.
Math Expert
Joined: 02 Sep 2009
Posts: 58133
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0  [#permalink]

### Show Tags

04 Sep 2019, 22:35
pzgupta wrote:
@Bunel - can you please explain how to approach this?
I am also confused of how to simplify the equation given in the question stem.

My solution is on the first page: https://gmatclub.com/forum/is-x-3-2-1-2 ... ml#p737280
_________________
Re: Is ((x-3)^2)^(1/2) = 3-x? (1) x ≠ 3 (2) -x|x| > 0   [#permalink] 04 Sep 2019, 22:35

Go to page   Previous    1   2   [ 28 posts ]

Display posts from previous: Sort by