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If x and y are integers such that (x+1)^2 less than equal to [#permalink]
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]
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.
Re: In equalities how to handle an expression with squares [#permalink]
12 Aug 2010, 12:01
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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.
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 \(|y-1|<{8}\) --> \({-7}<{y}<{9}\).
Re: In equalities how to handle an expression with squares [#permalink]
28 Aug 2012, 03:10
1
This post received KUDOS
Expert's post
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 \({-6}\leq{y}\leq{8}\).
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]
05 Dec 2012, 04:06
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]
22 Jul 2013, 08:37
2
This post received KUDOS
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]
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]
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]
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?
Re: If x and y are integers such that (x+1)^2 less than equal to [#permalink]
06 Dec 2014, 04:54
Expert's post
ikishan wrote:
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?
Isn't it explained in the very post you are quoting? _________________
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