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Inequality and absolute value questions from my collection

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Inequality and absolute value questions from my collection [#permalink] New post 16 Nov 2009, 10:33
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Guys I didn't forget your request, just was collecting good questions to post.

So here are some inequality and absolute value questions from my collection. Not every problem below is hard, but there are a few, which are quite tricky. Please provide your explanations along with the answers.

1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-20.html#p653690

2. If y is an integer and y = |x| + x, is y = 0?
(1) x < 0
(2) y < 1

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-20.html#p653695

3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653697

4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653709

5. What is the value of y?
(1) 3|x^2 -4| = y - 2
(2) |3 - y| = 11

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653731

6. If x and y are integer, is y > 0?
(1) x +1 > 0
(2) xy > 0

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653740

7. |x+2|=|y+2| what is the value of x+y?
(1) xy<0
(2) x>2 y<2

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653783 AND inequality-and-absolute-value-questions-from-my-collection-86939-160.html#p1111747

8. a*b#0. Is |a|/|b|=a/b?
(1) |a*b|=a*b
(2) |a|/|b|=|a/b|

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653789

9. Is n<0?
(1) -n=|-n|
(2) n^2=16

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653792

10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653796

11. Is |x+y|>|x-y|?
(1) |x| > |y|
(2) |x-y| < |x|

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653853

12. Is r=s?
(1) -s<=r<=s
(2) |r|>=s

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653870

13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653886

Official answers (OA's) and detailed solutions are in my posts on pages 2 and 3.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 05 Feb 2011, 08:35
Expert's post
ajit257 wrote:
Bunuel wrote:
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.


Bunuel...here why cant we use squaring on both sides of the statements.


I done't understand what you mean. Can you please SHOW what you mean.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 06 Feb 2011, 14:11
nnnnnnnniiiiiiiceeeeeee!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Feb 2011, 16:26
Bunuel wrote:
ajit257 wrote:
Bunuel wrote:
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.


Bunuel...here why cant we use squaring on both sides of the statements.


I done't understand what you mean. Can you please SHOW what you mean.


well i was just wanted to know why cant we do this

1. x + y = 3sqrt a
2. x - y = sqrt a

then solve the 2 equations to get x and y.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Feb 2011, 16:37
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Feb 2011, 16:47
oh yes ....yup got it ...thanks
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Feb 2011, 18:06
lagomez wrote:
Bunuel wrote:
5. What is the value of y?
(1) 3|x^2 -4| = y - 2
(2) |3 - y| = 11

(1) As we are asked to find the value of y, from this statement we can conclude only that y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative. Not sufficient.

(2) |3 - y| = 11:

y<3 --> 3-y=11 --> y=-8
y>=3 --> -3+y=11 --> y=14

Two values for y. Not sufficient.

(1)+(2) y>=2, hence y=14. Sufficient.

Answer: C.


So here in statement 1 we are not at all concerned with |X^2 -4|
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Feb 2011, 03:33
Expert's post
ajit257 wrote:
lagomez wrote:
Bunuel wrote:
5. What is the value of y?
(1) 3|x^2 -4| = y - 2
(2) |3 - y| = 11

(1) As we are asked to find the value of y, from this statement we can conclude only that y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative. Not sufficient.

(2) |3 - y| = 11:

y<3 --> 3-y=11 --> y=-8
y>=3 --> -3+y=11 --> y=14

Two values for y. Not sufficient.

(1)+(2) y>=2, hence y=14. Sufficient.

Answer: C.


So here in statement 1 we are not at all concerned with |X^2 -4|


Yes. We just have that left hand side is some absolute value and as absolute value is always non-negative then right hand side must also be non-negative.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 22 Feb 2011, 05:24
This is amazing. Thanks!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 22 Feb 2011, 05:30
Those are very tough. Couldn't finish many of them.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 02 Mar 2011, 09:13
This was good Thanks alot for the explanations
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Re: Inequality and absolute value questions from my collection [#permalink] New post 18 Jul 2011, 13:41
absolute brilliance :lol: from Bunuel!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 02 Aug 2011, 02:11
Nice questions. Thanks for sharing
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Aug 2011, 10:20
Quote:
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0
First, devide the whole equation by y, and
6*x=x^2+9 => x^2-6*x+9=0 => (x-3)^2=0, x=3, -3
(1) We know x could be 3 oro -3, let's put these two numbers in and see, if x=3, y =6, xy= 18,... and if x=-3, y =0, xy= 0. We can't know for sure what the value of that is. .... Insufficient
(2)Now we know X^3 is less than 0, then x must be less than 0 too. that means it has to be -3, then the answer comes out! .... Sufficient
B


Thanks ImJun for the above solution.

I just could not understand how option B was sufficient. I had left out on an important bit, (x-3)^2 will have both positive and negative roots.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Aug 2011, 10:39
Bunuel wrote:
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x + y)^2 = 9a --> x^2+2xy+y^2=9a. Clearly insufficient.

(2) (x – y)^2 = a --> x^2-2xy+y^2=a. Clearly insufficient.

(1)+(2) Add them up 2(x^2+y^2)=10a --> x^2+y^2=5a. Also insufficient as x,y, and a could be 0 and x^2 + y^2 > 4a won't be true, as LHS and RHS would be in that case equal to zero. Not sufficient.

Answer: E.



duh! missed out on the 0 and thought C was the ans.

Is this what you call the ZIP trap.. if so, then i sure did get zipped.. dang
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Aug 2011, 18:53
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0
First, devide the whole equation by y, and
6*x=x^2+9 => x^2-6*x+9=0 => (x-3)^2=0, x=3, -3
(1) We know x could be 3 oro -3, let's put these two numbers in and see, if x=3, y =6, xy= 18,... and if x=-3, y =0, xy= 0. We can't know for sure what the value of that is. .... Insufficient
(2)Now we know X^3 is less than 0, then x must be less than 0 too. that means it has to be -3, then the answer comes out! .... Sufficient
B

There is only one answer from the equation that you formed above from the question stem.
x^2 - 6xy +9 = 0

X =3

There for statement 1 is sufficient

Answer is A
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Aug 2011, 20:54
manishgeorge wrote:
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0
First, devide the whole equation by y, and
6*x=x^2+9 => x^2-6*x+9=0 => (x-3)^2=0, x=3, -3
(1) We know x could be 3 oro -3, let's put these two numbers in and see, if x=3, y =6, xy= 18,... and if x=-3, y =0, xy= 0. We can't know for sure what the value of that is. .... Insufficient
(2)Now we know X^3 is less than 0, then x must be less than 0 too. that means it has to be -3, then the answer comes out! .... Sufficient
B

There is only one answer from the equation that you formed above from the question stem.
x^2 - 6xy +9 = 0

X =3

There for statement 1 is sufficient

Answer is A


read Bunuel's and ImJun's post for this problem.

He writes that we can't divide both sides by 'y' since that would mean that we are assuming that y is not equal to 0.

The question prompt asks us the value of xy

Sol. would be:

If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

6*x*y = x^2*y + 9*y
x^2*y - 6*x*y + 9*y = 0
y (x^2 - 6*x + 9) = 0
y (x - 3)^2 = 0

Thus y could be = 0 or some other no.
X = -3 and +3

(1) y – x = 3

If x = 3, then y = 6
x = -3, then y = 0

thus two solutions, xy = 18 and xy = 0

Insuff

(2) x^3< 0

if x^3 < 0, we know that x = -3. Thus y = 0

Ans: xy = 0

Stmt 2 suff

Ans B

Hope this helps.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 08 Aug 2011, 21:12
I understand now. Thanks for explaining
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Re: Inequality and absolute value questions from my collection [#permalink] New post 15 Aug 2011, 05:09
great effort..thanks
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Re: Inequality and absolute value questions from my collection [#permalink] New post 25 Aug 2011, 20:53
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

Ans:

From the initial equation we get: y*(x^2-6x+9)=0 i.e y=0 or (x-3)^2=0
means either y=0 or (x-3)^2=0 or both
A does not give any idea about
B states that x^3<0 which means that x is not equal to 3.hence in that case y needs to be zero. so statement 2 itself is sufficient to answer the question.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Sep 2011, 20:47
1. If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

Hi Bunuel,

Different approach and I got the answer C. Please correct if I am wrong!

6*x*y= x^2*y + 9*y

(1) y - x = 3 => y = x+3
Substitue y into stem, we have:
6*x*(x+3) = x^2*(x+3) + 9*(x+3)
6*x^2 + 18*x = x^3 + 3*x^2 + 9*x +27
x^3 - 3*x^2 - 9*x + 27 = 0
(x-3)^2*(x+3) = 0
x = 3 or x = -3 (Insufficient)

(2) x^3 <0

(1) + (2), we have x = -3, y =0, then x*y = 0

Then the answer must be C
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Re: Inequality and absolute value questions from my collection   [#permalink] 07 Sep 2011, 20:47
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