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Quant Review #152: Inequalities

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by tonebeeze » Fri Jan 14, 2011 4:25 pm
152.

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?

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

OA = 13
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by stormier » Fri Jan 14, 2011 4:43 pm
tonebeeze wrote:152.

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?

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


x = (12-y)/2

For x to be an integer, (12-y) must be even.
Thus y must be even.
Since y lies between -12 and +12 (both inclusive) there are a total of 13 even numbers possible.
(-12,-10,-8,-6,-4,-2,0,2,4,6,8,10,12)

Thus there can be 13 possible solutions such that both x and y are integers.

OA = 13
Last edited by stormier on Fri Jan 14, 2011 4:45 pm, edited 1 time in total.
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by GMATGuruNY » Fri Jan 14, 2011 4:45 pm
tonebeeze wrote:152.

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?

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

OA = 13
Since 2x = even integer, y must also be an even integer.
Since |y|≤12, y can be any even integer from -12 to 12.

Here is the formula for counting even integers between one even integer and another:

(Biggest - Smallest)/2 + 1

Thus, the number of even integers from -12 to 12 = (12-(-12))/2 + 1 = 13.

The correct answer is D.
Last edited by GMATGuruNY on Sat Jan 15, 2011 3:55 am, edited 1 time in total.
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by bomond » Fri Jan 14, 2011 9:11 pm
GMATGuruNY wrote:
tonebeeze wrote:152.


The formula for counting even integers is:

(Biggest - Smallest)/2 + 1
Hi. I just want to say that that formula works only to integers that begins and ends with even numbers.
If the set begins with odd numbers then it will be a little bit different
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by Night reader » Fri Jan 14, 2011 11:16 pm
tonebeeze wrote:152.

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?

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

OA = 13
Hi tonebeeze, legs up and hands down :)

2x + y = 12 OR y=12-2x
|y| <= 12

Cool - now |12-2x| <=12 and
a) 12-2x <=12
b) 2x-12 <=12

a) x >= 0
b) deleted (x>=12) sorry mechanical mistake 2x-12<=12, x<=12
solution interval is below
0,1,2,3,4,5,6,7,8,9,10,11,12 => all in all we got 13 integers
Last edited by Night reader on Sat Jan 15, 2011 12:39 pm, edited 1 time in total.
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by ankur.agrawal » Fri Jan 14, 2011 11:58 pm
Night reader wrote:
tonebeeze wrote:152.

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?

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

OA = 13
Hi tonebeeze, legs up and hands down :)

2x + y = 12 OR y=12-2x
|y| <= 12

Cool - now |12-2x| <=12 and
a) 12-2x <=12
b) 2x-12 <=12

a) x >= 0
b) x>=12

Can I ask you a question ;) how we know that our interval starts at 0 and goes through exactly 12? - you must answer we know for sure ... how about going through 13 too - you may answer (dunnow, may be, no, yes, feel tired :) )
in fact x>= 12 and x can be 12 1/2 too :( so cancel the values beyond 12

0,1,2,3,4,5,6,7,8,9,10,11,12 => all in all we got 13 integers, correct?
Hey 2x-12<=12 implies 2x<=24 i.e. dividing by two both sides gives us x<=12.

How do u get x>=12. Plz clarify.

The rule as i know is multiplying or dividing by a negative no flips the sign
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by GMATGuruNY » Sat Jan 15, 2011 3:52 am
bomond wrote:
GMATGuruNY wrote:
tonebeeze wrote:152.


The formula for counting even integers is:

(Biggest - Smallest)/2 + 1
Hi. I just want to say that that formula works only to integers that begins and ends with even numbers.
If the set begins with odd numbers then it will be a little bit different
Correct! When I'm counting even integers, I begin and end the set with even values and then apply the formula. So to count the even integers between 27 and 53, I would apply the formula to the range of integers from 28 to 52:

Number of even integers between 28 and 52 inclusive = (52-28)/2 + 1 = 13.

I use the same process to count odd integers: I begin and end the set with odd values and then apply the formula. To count the odd integers between -12 and 12, I would apply the formula to the range of integers from -11 to 11:

Number of odd integers between -11 and 11 inclusive = 11-(-11)/2 + 1 = 12.
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by Night reader » Sat Jan 15, 2011 12:33 pm
ankur.agrawal wrote:
Night reader wrote:
tonebeeze wrote:152.

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?

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

OA = 13
Hi tonebeeze, legs up and hands down :)

2x + y = 12 OR y=12-2x
|y| <= 12

Cool - now |12-2x| <=12 and
a) 12-2x <=12
b) 2x-12 <=12

a) x >= 0
b) x>=12

Can I ask you a question ;) how we know that our interval starts at 0 and goes through exactly 12? - you must answer we know for sure ... how about going through 13 too - you may answer (dunnow, may be, no, yes, feel tired :) )
in fact x>= 12 and x can be 12 1/2 too :( so cancel the values beyond 12

0,1,2,3,4,5,6,7,8,9,10,11,12 => all in all we got 13 integers, correct?
Hey 2x-12<=12 implies 2x<=24 i.e. dividing by two both sides gives us x<=12.

How do u get x>=12. Plz clarify.

The rule as i know is multiplying or dividing by a negative no flips the sign
Ankur thanks for your note. I made mechanical mistake there.
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