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

 It is currently 17 Oct 2019, 13:27

### 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

# If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y

Author Message
TAGS:

### Hide Tags

Senior Manager
Joined: 01 Aug 2008
Posts: 458
If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

### Show Tags

09 Jan 2010, 14:17
4
41
00:00

Difficulty:

85% (hard)

Question Stats:

47% (01:37) correct 53% (02:13) wrong based on 490 sessions

### HideShow timer Statistics

If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$

(2) $$|x - 3|\leq{-y}$$
Math Expert
Joined: 02 Sep 2009
Posts: 58402
If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

### Show Tags

21 Jan 2012, 09:23
19
12
If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$. As given that $$y$$ is non negative value then $$|x - 3|$$ is more than (or equal to) some non negative value, (we could say the same ourselves as absolute value in our case ($$|x - 3|$$) is never negative). So we can not determine single numerical value of $$x$$. Not sufficient.

Or another way: to check $$|x - 3|\geq{y}\geq{0}$$ is sufficient or not just plug numbers:
A. $$x=5$$, $$y=1>0$$, and B. $$x=8$$, $$y=2>0$$: you'll see that both fits in $$|x - 3|>=y$$, $$y\geq{0}$$.

Or another way:
$$|x - 3|\geq{y}$$ means that:

$$x - 3\geq{y}\geq{0}$$ when $$x-3>0$$ --> $$x>3$$

OR (not and)
$$-x+3\geq{y}\geq{0}$$ when $$x-3<0$$ --> $$x<3$$

Generally speaking $$|x - 3|\geq{y}\geq{0}$$ means that $$|x - 3|$$, an absolute value, is not negative. So, there's no way you'll get a unique value for $$x$$. INSUFFICIENT.

(2) $$|x-3|\leq{-y}$$. Now, as $$|x-3|$$ is never negative ($$0\leq{|x-3|}$$) then $$0\leq{-y}$$ --> $$y\leq{0}$$ BUT stem says that $$y\geq{0}$$ thus $$y=0$$. $$|x-3|\leq{0}$$ --> $$|x-3|=0=y$$ (as absolute value, in our case |x-3|, can not be less than zero) --> $$x-3=0$$ --> $$x=3$$. SUFFICIENT

In other words:
$$-y$$ is zero or less, and the absolute value ($$|x-3|$$) must be at zero or below this value. But absolute value (in this case $$|x-3|$$) can not be less than zero, so it must be $$0$$.

Hope it helps..
_________________
##### General Discussion
SVP
Status: Nothing comes easy: neither do I want.
Joined: 12 Oct 2009
Posts: 2492
Location: Malaysia
Concentration: Technology, Entrepreneurship
Schools: ISB '15 (M)
GMAT 1: 670 Q49 V31
GMAT 2: 710 Q50 V35

### Show Tags

08 May 2010, 10:35
2
neoreaves wrote:
If y > = 0, what is the value of x?
1. |x - 3| >= y
2. |x - 3| <= - y

IMO B

Statement 1). |x - 3| >= y >=0
|x - 3| >= 0 , for different values of x, this is true.

Statement 2). |x - 3| <= - y since |x - 3| is always >=0 , and y>=0

|x - 3| <= - y will hold true only when y is 0

=> |x - 3| = 0 , only solution is x=3 hence sufficient.

Thus B
_________________
Fight for your dreams :For all those who fear from Verbal- lets give it a fight

Money Saved is the Money Earned

Jo Bole So Nihaal , Sat Shri Akaal

GMAT Club Premium Membership - big benefits and savings

Gmat test review :
http://gmatclub.com/forum/670-to-710-a-long-journey-without-destination-still-happy-141642.html
Intern
Joined: 30 May 2012
Posts: 18
Concentration: Finance, Strategy
GMAT 1: 730 Q49 V41
GPA: 3.39
If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

### Show Tags

09 Jul 2013, 14:05

First number line: y>=0
Second: From properties of absolute value, |x-3|>=0 is true no matter what x is.
Third: (2) tells us that |x-3|<=-y. We know nothing about y other than y>=0. Therefore |x-3|<=-y only tells us that |x-3|<=0.
Fourth: Since |x-3|<=0 AND |x-3|>=0, we now know the exact value of |x-3|. In the diagram it is the point on the number line that is both <=0 and >=0. The only value for which that is true is 0. So |x-3|=0. From there you know that x-3=0 and so x=3.
Senior Manager
Joined: 13 May 2013
Posts: 399
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

10 Jul 2013, 13:43
If y >= 0, What is the value of x?

(1) |x-3| >= y
We are told that |x-3| ≥ y but all we know is that y is a positive # greater than or equal to zero. Therefore, all we know is that |x-3| is greater than or equal to zero and x could be an infinite number of possibilities.

(2) |x-3| <= -y

(I am having real difficulty understanding the rationale for (2) could someone explain it to me? (preferably like they would to a 5th grader )

Thanks!
Retired Moderator
Joined: 16 Jun 2012
Posts: 988
Location: United States
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

10 Jul 2013, 14:58
2
WholeLottaLove wrote:
If y >= 0, What is the value of x?

(1) |x-3| >= y
We are told that |x-3| ≥ y but all we know is that y is a positive # greater than or equal to zero. Therefore, all we know is that |x-3| is greater than or equal to zero and x could be an infinite number of possibilities.

(2) |x-3| <= -y

(I am having real difficulty understanding the rationale for (2) could someone explain it to me? (preferably like they would to a 5th grader )

Thanks!

Hi WholeLottaLove

The important property concerning absolute value inequalities is:
|a| <= b <--> -b <= a <=b
[a is in the middle of -b and b, inclusive]

Apply to statement (2)
|x - 3| <= -y
<--> -(-y) <= (x-3) <= -y,
<--> y <= (x-3) <= -y

We know y is 0 or positive, -y is 0 or negative.
Because there is not any number that is both negative and positive.Thus, there is ONLY one number that is both >= y AND <= -y. That is zero.
Therefore, (x-3) must be zero. ==> x = 3

Hope it's clear.
_________________
Please +1 KUDO if my post helps. Thank you.

"Designing cars consumes you; it has a hold on your spirit which is incredibly powerful. It's not something you can do part time, you have do it with all your heart and soul or you're going to get it wrong."

Chris Bangle - Former BMW Chief of Design.
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 15267
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: If y >= 0, what is the value of x?  [#permalink]

### Show Tags

16 May 2015, 11:43
1
Hi All,

When complex-looking questions show up on Test Day, there's almost always some type of built-in pattern involved. If you can't immediate spot the pattern, then you have to put in a bit of work to prove what the pattern actually is....TESTing VALUES can help you to prove that a pattern exists.....

Here, we're told that Y >= 0. We're asked for the value of X.

Fact 1: |X-3| >= Y

IF....
Y = 0
Then |X-3| >= 0, so X can be ANY number. As Y gets bigger, certain options are eliminated, but given this 'restriction', X has an infinite number of possibilities.
Fact 1 is INSUFFICIENT

Fact 2: |X-3| <= -Y

Here, we have to be CAREFUL with the details. Notice how there's a NEGATIVE sign in front of the Y.....

IF....
Y = 0
|X-3| <= 0

Absolute values CANNOT have negative results - the result is ALWAYS 0 or a positive, so this TEST has JUST ONE solution...
X = 3

IF....
Y = 1
|X-3| <= - 1 which is NOT POSSIBLE.

From the prompt, we know that Y >= 0, so choosing a positive value for Y will NOT fit the absolute value given in Fact 2. This means that the ONLY possible value for Y is 0. By extension, there is ONLY ONE possible value for X....X = 3.
Fact 2 is SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
Intern
Joined: 05 Jun 2015
Posts: 24
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

06 Mar 2016, 19:57
Bunuel wrote:

There is a following problem with your solution:
If $$x<3$$ --> $$-(x-3)\geq{y}$$ --> $$3-y\geq{x}$$;
OR:
If $$x\geq{3}$$ --> $$(x-3)\geq{y}$$ --> $$x\geq{3+y}$$;

But you can not combine these inequalities and write: $$3+y\leq{x}\leq{3-y}$$ as they are OR scenarios not AND scenarios (meaning that depending on the value of x we'll have either the first one or the second one).

Also discussed here: if-y-geq-0-what-is-the-value-of-x-1-x-3-geq-y-91640.html

Hope it helps..

Hi Bunuel,

I understand the OR scenario you mentioned above. However, what if the statement reads $$|x-3|\leq{y}$$? Then we can have $$3-y\leq{x}\leq{3+y}$$, can't we? So here we have the AND scenario. Is that right?

And if I am correct, how can we solve the above inequalities, given that $$y\geq{0}$$?

Thank you very much!
Math Expert
Joined: 02 Aug 2009
Posts: 7971
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

06 Mar 2016, 20:51
1
truongynhi wrote:
Bunuel wrote:

There is a following problem with your solution:
If $$x<3$$ --> $$-(x-3)\geq{y}$$ --> $$3-y\geq{x}$$;
OR:
If $$x\geq{3}$$ --> $$(x-3)\geq{y}$$ --> $$x\geq{3+y}$$;

But you can not combine these inequalities and write: $$3+y\leq{x}\leq{3-y}$$ as they are OR scenarios not AND scenarios (meaning that depending on the value of x we'll have either the first one or the second one).

Also discussed here: if-y-geq-0-what-is-the-value-of-x-1-x-3-geq-y-91640.html

Hope it helps..

Hi Bunuel,

I understand the OR scenario you mentioned above. However, what if the statement reads $$|x-3|\leq{y}$$? Then we can have $$3-y\leq{x}\leq{3+y}$$, can't we? So here we have the AND scenario. Is that right?

And if I am correct, how can we solve the above inequalities, given that $$y\geq{0}$$?

Thank you very much!

Hi,
this too is OR scenario,
because $$3-y\leq{x}$$ and $${x}\leq{3+y}$$ are dependent on different set of values of x..
$$3-y\leq{x}$$ is when $$x<3$$..
and $${x}\leq{3+y}$$ is when $$x\geq{3}$$..
Example of AND is
when we have two eqs in x, and not dependent on each other..
say x<3.. and x+2>1..
_________________
Intern
Joined: 05 Jun 2015
Posts: 24
Location: Viet Nam
GMAT 1: 740 Q49 V41
GPA: 3.66
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

06 Mar 2016, 22:08
Hi chetan2u,

Thank you for the prompt reply. But I still have a doubt.

Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$. What I understand is that x must be greater than -2 AND less than 2 for the inequalitiy to hold. So I think this is an AND scenario.

If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario. Here OR makes sense to me.

My inequality skill is pretty rusty. Thank you for bearing with me. I very appreciate your help!
Math Expert
Joined: 02 Aug 2009
Posts: 7971
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

06 Mar 2016, 22:26
truongynhi wrote:
Hi chetan2u,

Thank you for the prompt reply. But I still have a doubt.

Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$. What I understand is that x must be greater than -2 AND less than 2 for the inequalitiy to hold. So I think this is an AND scenario.

If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario. Here OR makes sense to me.

My inequality skill is pretty rusty. Thank you for bearing with me. I very appreciate your help!

Hi truongynhi,

I am happy to help you and clear a few doubts you have..

WHAt does OR and AND mean..

1) Take the inequation $$x^2<4$$ for example. We then have $$|x|<2$$, which means $$-2<x<2$$
So here too you had 2 inequalities, x<2 and x>-2..
x<2 can mean x is -3 so this is a solution when we are using OR since we are not looking at both together..
But here x<2 and x>-2 has a range which OVERLAPS, so this is the combined solution for two inequalities..
when you are choosing a value in this range, you are using AND, since taht value will satisfy both the inequalities..

2)If, however, $$x^2>4$$, then $$x<-2$$ OR $$x>2$$. This is clearly an OR scenario.
YES, this is OR situation, because there is no overlap and hence there is no possiblity of a combined solution..
any solution will satisfy just one inequality..

Now when you have two variable as in the case of Q mentioned here..
3+y≤x≤3−y.. you cannot take this as a solution..
WHY..
because the two inequalities you have combined have come in two different scenarios of OR while taking value of x..
there is no OVERLAP in values of x.. x<3 for one inequality and x>= 3 for other..
so don't combine the two..

_________________
GMAT Tutor
Joined: 24 Jun 2008
Posts: 1806
Re: If y>=0, What is the value of x?  [#permalink]

### Show Tags

06 May 2019, 07:53
Apex231 wrote:
If y >= 0, What is the value of x?

(1) |x-3| >= y
(2) |x-3| <= -y

Statement 1 is not sufficient, since if y=0, x can be anything.

In Statement 2, the left side is 0 or greater, since it is an absolute value. The right side is 0 or smaller, since it is the negative of y, which is at least zero. If the left side (which is > 0) is no bigger than the right side (which is < 0), the only possibility is that they are both exactly equal to zero. And that only happens if x=3.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
SVP
Joined: 03 Jun 2019
Posts: 1699
Location: India
Re: If y>=0, What is the value of x? (1) |x-3|>=y (2) |x-3|<=-y  [#permalink]

### Show Tags

24 Sep 2019, 09:24
ugimba wrote:
If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$

(2) $$|x - 3|\leq{-y}$$

If $$y\geq{0}$$, what is the value of x?

(1) $$|x - 3|\geq{y}$$
$$|x - 3|\geq{y}\geq{0}$$
$$|x-3| \geq 0$$
NOT SUFFICIENT

(2) $$|x - 3|\leq{-y}$$
$$|x - 3|\leq{-y} \leq 0$$
$$|x-3| \leq 0$$
|x-3| = 0
x = 3
SUFFICIENT

IMO B
_________________
"Success is not final; failure is not fatal: It is the courage to continue that counts."

Please provide kudos if you like my post. Kudos encourage active discussions.

My GMAT Resources: -

Efficient Learning
All you need to know about GMAT quant

Tele: +91-11-40396815
Mobile : +91-9910661622
E-mail : kinshook.chaturvedi@gmail.com
Non-Human User
Joined: 09 Sep 2013
Posts: 13243
Re: If y is greater than or equal to 0, what is the value of x?  [#permalink]

### Show Tags

17 Oct 2019, 04:38
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________
Re: If y is greater than or equal to 0, what is the value of x?   [#permalink] 17 Oct 2019, 04:38
Display posts from previous: Sort by