Find all School-related info fast with the new School-Specific MBA Forum

 It is currently 21 Oct 2016, 08:32

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# For how many ordered pairs (x, y) that are solutions of the

Author Message
TAGS:

### Hide Tags

Manager
Status: Current MBA Student
Joined: 19 Nov 2009
Posts: 127
Concentration: Finance, General Management
GMAT 1: 720 Q49 V40
Followers: 13

Kudos [?]: 316 [3] , given: 210

For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

14 Jan 2011, 16:40
3
KUDOS
25
This post was
BOOKMARKED
00:00

Difficulty:

55% (hard)

Question Stats:

60% (02:23) correct 40% (01:29) wrong based on 1125 sessions

### HideShow timer Statistics

$$2x + y = 12$$
$$|y| \leq 12$$

For how many ordered pairs (x, y) that are solutions of the system above are x and y both integers?

A. 7
B. 10
C. 12
D. 13
E. 14
[Reveal] Spoiler: OA
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6962
Location: Pune, India
Followers: 2024

Kudos [?]: 12711 [14] , given: 221

### Show Tags

14 Jan 2011, 17:28
14
KUDOS
Expert's post
5
This post was
BOOKMARKED
tonebeeze wrote:
152.

$$2x + y = 12$$
$$|y| \leq 12$$

For how many ordered pairs (x, y) that are solutions of the system above are x and y both integers?

a. 7
b. 10
c. 12
d. 13
e. 14

The solution of $$|y| \leq 12$$ is straight forward.
$$-12 \leq y \leq 12$$
(If you are not comfortable with this, check out my blog post:
http://www.veritasprep.com/blog/2011/01/quarter-wit-quarter-wisdom-do-what-dumbledore-did/

If both x and y have to be integers, y should be an integer and hence can take any value from the set {-12, -11, -10 ... 10, 11, 12} i.e. any one of 25 values (these are 25 values -12 to -1 (12 values), 0, 1 to 12 (another 12 values)) 13 of them are even and 12 of them are odd.

$$2x + y = 12$$
Every time y is even, x will be integer. e.g. y = 12, x = 0 (because x = (12 - even)/2 will be an integer)
Every time y is odd, x will be non-integer e.g. y = 1, x = 5.5 (because x = (12 - odd)/2 will not be an integer)

Therefore, for 13 values, x and y both will be integers.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Math Forum Moderator Joined: 20 Dec 2010 Posts: 2021 Followers: 158 Kudos [?]: 1618 [3] , given: 376 Re: 152. Algebra Absolute value [#permalink] ### Show Tags 10 Mar 2011, 11:15 3 This post received KUDOS Sol: |y| <= 12 Means; -12<=y<=12 2x + y = 12 x = (12-y)/2 x will be integers for y=even; because even-even = even and even is always divisible by 2. We need to find out how many even integers are there between -12 and 12 ((12-(-12))/2)+1 = (24/2)+1 = 12+1 = 13 Ans: "D" _________________ Math Expert Joined: 02 Sep 2009 Posts: 35240 Followers: 6619 Kudos [?]: 85310 [5] , given: 10236 Re: 152. Algebra Absolute value [#permalink] ### Show Tags 10 Mar 2011, 11:19 5 This post received KUDOS Expert's post 1 This post was BOOKMARKED Baten80 wrote: 2x + y = 12 |y| <= 12 152. For how many ordered pairs (x , y) that are solutions of the system above are x and y both integers? A. 7 B. 10 C. 12 D. 13 E. 14 Given: $$-12\leq{y}\leq{12}$$ and $$2x+y=12$$ --> $$y=12-2x=2(6-x)=even$$, (as $$x$$ must be an integer). Now, there are 13 even numbers in the range from -12 to 12, inclusive each of which will give an integer value of $$x$$. Answer: D. _________________ VP Status: There is always something new !! Affiliations: PMI,QAI Global,eXampleCG Joined: 08 May 2009 Posts: 1353 Followers: 17 Kudos [?]: 230 [1] , given: 10 Re: 152. Algebra Absolute value [#permalink] ### Show Tags 16 Jun 2011, 00:00 1 This post received KUDOS -12<= y <=12 gives 0<=x <=12 thus 13 values in total. _________________ Visit -- http://www.sustainable-sphere.com/ Promote Green Business,Sustainable Living and Green Earth !! Manager Joined: 26 Jul 2011 Posts: 125 Location: India WE: Marketing (Manufacturing) Followers: 1 Kudos [?]: 97 [0], given: 15 Re: Quant Rev. #152 [#permalink] ### Show Tags 07 Sep 2012, 00:22 Hi Karishma Using the number properties this indeed is very convenient to solve. I was wondering can we substitute y = 12 - 2x in the inequality and solve for the possible values of x. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 6962 Location: Pune, India Followers: 2024 Kudos [?]: 12711 [2] , given: 221 Re: Quant Rev. #152 [#permalink] ### Show Tags 09 Sep 2012, 21:21 2 This post received KUDOS Expert's post 1 This post was BOOKMARKED ratinarace wrote: Hi Karishma Using the number properties this indeed is very convenient to solve. I was wondering can we substitute y = 12 - 2x in the inequality and solve for the possible values of x. Certainly and it is quick too. y = 12 - 2x Whenever x is an integer, y will be an integer. So if we can solve for integral values of x, the number of values we get will be the number of solutions. $$|y| \leq 12$$ $$|12 - 2x| \leq 12$$ $$|x - 6| \leq 6$$ From 6, x should be at a distance less than or equal to 6. So x will lie from 0 to 12 i.e. 13 values. (Check http://www.veritasprep.com/blog/2011/01 ... edore-did/ if this is not clear) There are 13 solutions. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Manager
Joined: 26 Jul 2011
Posts: 125
Location: India
WE: Marketing (Manufacturing)
Followers: 1

Kudos [?]: 97 [0], given: 15

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

10 Sep 2012, 10:36
Thanks Karishma..Wonderful explaination
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 6962
Location: Pune, India
Followers: 2024

Kudos [?]: 12711 [0], given: 221

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

25 Sep 2012, 02:50
Responding to a pm:
Changing the sign within the mod has no impact on anything outside the mod.

$$|6 - x| \leq 12$$ is same as
$$|x - 6| \leq 12$$

Think about it: Whether you write |x| or |-x|, it is the same.
|6| = |-6|

So for every value of x,
|x - 6| = |6 - x|
So you don't need to flip the inequality sign.

|x - 6| and - |x - 6| are of course different. If you change |x - 6| to - |x - 6|, you will need to flip the inequality sign.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 35240
Followers: 6619

Kudos [?]: 85310 [0], given: 10236

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

04 Jul 2013, 01:44
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

_________________
Intern
Joined: 05 May 2013
Posts: 27
GMAT 1: 730 Q50 V39
GRE 1: 1480 Q800 V680
Followers: 0

Kudos [?]: 22 [3] , given: 5

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

05 Jul 2013, 03:54
3
KUDOS
$$y=12-2x=2*(6-x).$$
Since $$|y| \leq 12 , -12 \leq y \leq 12$$ . Substituting for y from above, $$-6 \leq (6-x) \leq 6.$$. This reduces to $$x \geq 0$$ and $$x \leq 12.$$ Including 0 and 12 there are thus 13 integer solutions.
Senior Manager
Joined: 13 May 2013
Posts: 472
Followers: 3

Kudos [?]: 146 [1] , given: 134

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

09 Jul 2013, 16:34
1
KUDOS
2x+y=12
|y|<=12

For how many ordered pairs (x, y) that are solutions of the system above are x and y both integers?

y=12-2x
|y|<=12
|12-2x| <= 12
12 - 2x <= 12
-2x <= 0
x>=0

-(12-2x) <= 12
-12+2x <= 12
2x <= 24
x<=12

13 solutions between 0 and 12 inclusive.
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 12153
Followers: 538

Kudos [?]: 151 [0], given: 0

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

08 Aug 2014, 07:41
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.
_________________
Intern
Joined: 17 May 2012
Posts: 49
Followers: 0

Kudos [?]: 9 [0], given: 126

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

24 Nov 2014, 22:20
Hi Moderators,

Does this qualify as a 700 level question, I think it should be in the lower range?

Thanks
Math Expert
Joined: 02 Sep 2009
Posts: 35240
Followers: 6619

Kudos [?]: 85310 [0], given: 10236

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

25 Nov 2014, 04:47
aj0809 wrote:
Hi Moderators,

Does this qualify as a 700 level question, I think it should be in the lower range?

Thanks

User statistics on the question say it's 700.
_________________
GMAT Club Legend
Joined: 09 Sep 2013
Posts: 12153
Followers: 538

Kudos [?]: 151 [0], given: 0

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

22 Dec 2015, 15:29
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.
_________________
Manager
Joined: 17 Nov 2013
Posts: 100
Followers: 0

Kudos [?]: 2 [0], given: 0

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

09 Apr 2016, 14:15
abs 12 = -12 all the way to 12, which is 25 integers including "0".

y= 2(6-x) = even.

So y = even. and y is equal the 25 number range. How many possible even numbers are in that range?

Answer is 13 possible even y numbers including zero "0".
Senior Manager
Joined: 22 Jun 2014
Posts: 454
Concentration: General Management, Technology
GMAT 1: 540 Q45 V20
GPA: 2.49
WE: Information Technology (Computer Software)
Followers: 12

Kudos [?]: 148 [0], given: 91

Re: For how many ordered pairs (x, y) that are solutions of the [#permalink]

### Show Tags

12 Apr 2016, 00:23
tonebeeze wrote:
$$2x + y = 12$$
$$|y| \leq 12$$

For how many ordered pairs (x, y) that are solutions of the system above are x and y both integers?

A. 7
B. 10
C. 12
D. 13
E. 14

|y| <= 12 means range of y is -12 <= Y <= +12. which means Y can take any of the value in the set (-12, -11, -10......-1,0,1.....10,11,12).

now that we are given 2x + y = 12, y = 12 - 2x

we can include all the integer values for X as a solution for y = 12 - 2x as long as y falls in the above range mentioned. Such values of X are (0,1,2....12). 13 is the count for this set. Answer is D.
_________________

---------------------------------------------------------------
Target - 720-740
helpful post means press '+1' for Kudos!
http://gmatclub.com/forum/information-on-new-gmat-esr-report-beta-221111.html
http://gmatclub.com/forum/list-of-one-year-full-time-mba-programs-222103.html

Re: For how many ordered pairs (x, y) that are solutions of the   [#permalink] 12 Apr 2016, 00:23
Similar topics Replies Last post
Similar
Topics:
4 How many solutions are possible for the inequality | x - 1 | + | x - 6 4 12 Sep 2016, 11:02
6 For how many integers pair (x,y) satisfies the result 5 18 Jan 2015, 15:01
35 For how many ordered pairs (x , y) that are solutions of the 4 11 Mar 2014, 23:54
2 For how many ordered pairs (x , y) that are solutions of the 7 17 Jul 2011, 00:05
27 For how many ordered pairs (x , y) that are solutions of the 8 24 May 2007, 06:07
Display posts from previous: Sort by