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If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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25 Jul 2010, 17:51

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If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy.

Re: In equalities how to handle an expression with squares [#permalink]

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25 Jul 2010, 17:58

gmatrant wrote:

If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy.

In equalities how to handle an expression which is squared Does the above equation (x+1)^2 <= 36 mean |x+1| < (+6 or -6)

I then get 4 equations.. and I am am not able to proceed. Can you someone please explain how such questions are to be handled.

If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy.

In equalities how to handle an expression which is squared Does the above equation (x+1)^2 <= 36 mean |x+1| < (+6 or -6)

I then get 4 equations.. and I am am not able to proceed. Can you someone please explain how such questions are to be handled.

If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy.

\((y-1)^2<{64}\) --> \({-\sqrt{64}}<{y-1}<{\sqrt{64}}\) --> \({-8}<{y-1}<{8}\) --> \({-7}<{y}<{9}\), as \(y\) is an integer we can rewrite this inequality as \({-6}\leq{y}\leq{8}\).

We should try extreme values of \(x\) and \(y\) to obtain min and max values of \(xy\):

Min possible value of \(xy\) is for \(x=-7\) and \(y=8\) --> \(xy=-56\); Max possible value of \(xy\) is for \(x=-7\) and \(y=-6\) --> \(xy=42\).

Solving with absolute values gives the same results:

\((x+1)^2\leq{36}\) means \(|x+1|\leq{6}\) --> \({-7}\leq{x}\leq{5}\). \((y-1)^2<{64}\) means \(|y-1|<{8}\) --> \({-7}<{y}<{9}\).

Hi Bunuel, " as y is an integer we can rewrite this inequality as -6<= y<=8 ."

I didn't understand as to how can we change the range of y from -7,9 to -6,8 ?

We are not changing the range here.

We have \({-7}<{y}<{9}\). Now, since \(y\) is an integer, then it can take integer values from -6 to 8, inclusive: -6, -5, ..., 6, 7, 8, which can be written as \({-6}\leq{y}\leq{8}\).

Now solving by the graphical approach, range is between (and inclusive of) -7 and 5.

Similarly solving for the other equation we get the roots in the range of (exclusive of) -7 and 9. Therefore maximum and minimum value of xy can be derived as others before me have explained.

Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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05 Dec 2012, 04:06

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One thing I notice is that you have to be careful with the exclusivity and inclusivity of ranges. In these questions, you will get it wrong if you thought y = 9 is included in the range.

\(|x+1| <= 6\) This means x is within the range of [-1-6,-1+6] = [-7,5]. This is inclusive of -7 and 5.

\(|y-1|<8\) This means y is within the range of (1-8,1+8) = (-7,9). This is exclusive of -7 and 9.

Now to get the extreme values.

Max value or Positive outcomes: -7 * -6 = 42 or 5 * 8 = 40 ==> 42 wins! Min value of Negative outcomes: -7 * 8 = -56 or -6 * 5 = -30 ==> -56 wins!

Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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22 Jul 2013, 08:37

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gmatrant wrote:

If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy.

Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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29 Jul 2013, 01:12

2. If x and y are integers such that (x+1)^2 less than equal to 36 and (y-1)^2 less than 64. What is the largest possible and minimum possible value of xy. (x+1)^2 < 36 (x+1) < +6 (applying square root on both sides) X < +6-1 X < 5

X >-6-1 X > -7 Range of possible values for x are between -7 and +5

(y-1)^2 < 64 (y-1) <+ 8 (applying square root on both sides) y < +8+1 y < +9

y < -8+1 y > -7 As y is an integer y < 8 or y > -6 Range of possible values for y are between -6 and +8

Range of possible values for xy at their respective highest and lowest levels: -30, 40, 42, -56 Highest value: 42 and lowest value: -56

Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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16 Sep 2014, 19:58

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Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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05 Dec 2014, 20:37

Bunuel wrote:

pallavisatsangi wrote:

Hi Bunuel, " as y is an integer we can rewrite this inequality as -6<= y<=8 ."

I didn't understand as to how can we change the range of y from -7,9 to -6,8 ?

We are not changing the range here.

We have \({-7}<{y}<{9}\). Now, since \(y\) is an integer, then it can take integer values from -6 to 8, inclusive: -6, -5, ..., 6, 7, 8, which can be written as \({-6}\leq{y}\leq{8}\).

Hope it's clear.

if it is asking for maximum and minimum value. why did you re-wrote it?

Hi Bunuel, " as y is an integer we can rewrite this inequality as -6<= y<=8 ."

I didn't understand as to how can we change the range of y from -7,9 to -6,8 ?

We are not changing the range here.

We have \({-7}<{y}<{9}\). Now, since \(y\) is an integer, then it can take integer values from -6 to 8, inclusive: -6, -5, ..., 6, 7, 8, which can be written as \({-6}\leq{y}\leq{8}\).

Hope it's clear.

if it is asking for maximum and minimum value. why did you re-wrote it?

Isn't it explained in the very post you are quoting?
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Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]

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02 Apr 2016, 00:29

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).

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