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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 Jul 2010, 14:23
this thread has helped a lot.. thanks!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 05 Jul 2010, 14:31
Whoa, inequalities and AV's did me in for my first GMAT! Great thread!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 06 Jul 2010, 23:47
Ive got C for this question..

when both together yields x^2 + y^2 = 5a
why it is E?

Also I dont understand the explanation of below,
St. (1) and (2) together : x^2 + y^2 = 5a
When either x or y is not 0, question stem holds true.
When x and y are both 0, question stem is false.

Can somebody clarify how to solve this please...


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


St. (1) : (x + y)^2 = 9a
x^2 + y^2 + 2xy = 9a
Insufficient.

St. (2) : (x - y)^2 = a
x^2 + y^2 - 2xy = a
Insufficient.

St. (1) and (2) together : x^2 + y^2 = 5a
When either x or y is not 0, question stem holds true.
When x and y are both 0, question stem is false.

Hence insufficient.

Answer : E
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Re: Inequality and absolute value questions from my collection [#permalink] New post 07 Jul 2010, 02:38
Expert's post
gmatJP wrote:
Ive got C for this question..

when both together yields x^2 + y^2 = 5a
why it is E?

Also I dont understand the explanation of below,
St. (1) and (2) together : x^2 + y^2 = 5a
When either x or y is not 0, question stem holds true.
When x and y are both 0, question stem is false.

Can somebody clarify how to solve this please...


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


St. (1) : (x + y)^2 = 9a
x^2 + y^2 + 2xy = 9a
Insufficient.

St. (2) : (x - y)^2 = a
x^2 + y^2 - 2xy = a
Insufficient.

St. (1) and (2) together : x^2 + y^2 = 5a
When either x or y is not 0, question stem holds true.
When x and y are both 0, question stem is false.

Hence insufficient.

Answer : E


OA' s and solutions for all the problems are given in my posts on pages 2 and 3.

OA for this question is E.

When we consider statement together we'll have: x^2+y^2=5a. Now, if x, y and a are different from zero (for example: x=3, y=4 and a=5) then x^2+y^2=5a>4a and the answer to the question is YES, but if x=y=a=0, then x^2+y^2=5a=4a=0 and the answer to the question is NO. Two different answers, thus not sufficient.

Hope it's clear.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Inequality and absolute value questions from my collection [#permalink] New post 16 Jul 2010, 07:03
Hi Bunuel,
If 6*x*y = x^2*y + 9*y, what is the value of xy?
(1) y – x = 3
(2) x^3< 0

y*(x-3)^2 = 0 means x=3 or/and y=0.

I have been thinking that ONLY either of the variables can be zero. i.e x=3 or y=0.
So should i always consider the possibility of both X=3 and Y=0 ?

Can you explain the concept?

Thanks
Vinod
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Re: Inequality and absolute value questions from my collection [#permalink] New post 16 Jul 2010, 07:07
Hi Bunuel,
I forgot to thank you for these great questions and solutions. You are the BEST!!!!

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

(2) y<1, as we concluded y is never negative, and we are given that y is an integer, hence . Sufficient.

In the above problem, it is very obvious that Y is not equal to 0 from the 2nd statement.
Do you think GMAT questions will have this much obvious statements?
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Re: Inequality and absolute value questions from my collection [#permalink] New post 16 Jul 2010, 07:21
Expert's post
ramadossvinodh wrote:
Hi Bunuel,
I forgot to thank you for these great questions and solutions. You are the BEST!!!!

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

(2) y<1, as we concluded y is never negative, and we are given that y is an integer, hence . Sufficient.

In the above problem, it is very obvious that Y is not equal to 0 from the 2nd statement.
Do you think GMAT questions will have this much obvious statements?


The trick here is to conclude that y can not be negative, the rest is relatively easy. And yes, I think you can see such questions on GMAT.
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COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Inequality and absolute value questions from my collection [#permalink] New post 20 Jul 2010, 01:34
lagomez wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

answer A
because in number 2 n can be negative or a fraction



Doubt :The question is is |N|<4 ,from statement 2 we can see that n can only be a fraction or its value lies between 0 and 1 like say 1/2 .1/3 etc.same for negative fractions
for example for n =1/2 , 1/(|1/2|)=2 which is greater than n=1/2

Also for n=-1/2,1/(|-1/2|)=2 which is greater than n=-1/2


for negative integers also its true ,say we take n=-3 ,
1/|n|=1/3 which is greater than n=-3



But point to note is that both negative nos and fractions are less than 4 ,so this statement is equally sufficient ,correct me if i am wrong !!!!
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Re: Inequality and absolute value questions from my collection [#permalink] New post 20 Jul 2010, 01:47
Expert's post
amlan009 wrote:
lagomez wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

answer A
because in number 2 n can be negative or a fraction



Doubt :The question is is |N|<4 ,from statement 2 we can see that n can only be a fraction or its value lies between 0 and 1 like say 1/2 .1/3 etc.same for negative fractions
for example for n =1/2 , 1/(|1/2|)=2 which is greater than n=1/2

Also for n=-1/2,1/(|-1/2|)=2 which is greater than n=-1/2


for negative integers also its true ,say we take n=-3 ,
1/|n|=1/3 which is greater than n=-3



But point to note is that both negative nos and fractions are less than 4 ,so this statement is equally sufficient ,correct me if i am wrong !!!!


\frac{1}{|n|}>n holds true for ALL negative values of n, as if n<0 then LHS=positive>RHS=negative. Hence we don't know whether -4<n<4 is true. That's why statement (2) is not sufficient.

The complete range of n for which \frac{1}{|n|}>n holds true is n<1.

P.S. OA's and solutions for all question are given in my posts on pages 2 and 3.

Hope it's clear.
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RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
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Re: Inequality and absolute value questions from my collection [#permalink] New post 22 Jul 2010, 05:36
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

2. If y is an integer and y = |x| + x, is y = 0?
(1) x < 0
(2) y < 1
(1)Sufficient
(2)Sufficient, y couldn't be negative, the least it could be is 0.
D
3. Is x^2 + y^2 > 4a?
(1) (x + y)^2 = 9a
(2) (x – y)^2 = a

(1) (x+y)^2= x^2+y^2+2x*y
9a=x^2+y^2+2x*y Still Insufficient

(2) (x-y)^2=x^2+y^2-2x*y
a=x^2+y^2-2x*y Insufficient

Sum up equations (1) and (2) , then we can get 10a=2(x^2+y^2) => 5a=x^2+y^2, and we know a is necessarily positive since (x – y)^2 = a
With both factors, we can know x^2 + y^2 >4a

the answer is C

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

(1)Insufficient
(2)that only tells you x is greater than y Insufficient
(1) and (2) together: x-y=0.5, and x is great than y, Insufficient
E


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

(1) we only know y must be positive, insufficient
(2)y coule be -8 or 14 insufficient
BOth (1) and (2): from statement (1) weve learned that y must be a positive number, along with statement (2) we know y is 14
the answer is C

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

(1)x>-1, we wouldnt know anything about y with that, insufficient
(2)xy>0, if x is less than 0, then y too has to be less than 0 Insufficient

both (1) and (2): xy>0, x>-1, x could be 0 or any positive integer, and even if x=0, y still has to be great than 0 coz xy>0 Sufficient

C

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

(1)xy<0, that means either x or y is positive and the other negative. seems like x+y= -4, coz x+2=-2-y => x+y=-4, sufficient
(2) Sufficient too, same reasons as above

D

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

(1) a*b has to be positive, a/b has to positive too. Sufficient

(2) we already know that for sure, Insufficient

A

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

(1)since the absolute value must be positive, so we know n has to be negative, or 0 Not Sufficient
(2)ok, n could be 4 or -4, so what? Insufficient
BOth (1) and (2) together: ok, n could be 4 or -4, and it's less than or equal to zero, so it must be -4 Sufficient

C

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

(1) n>4 or n<-4, |n| must be greater than 4 sufficient
(2)we only know n<1, |n| could be greater than 4 if n < -4 insufficient

A

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

Just leave it for a while


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

(1) Not sufficient
(2) Not sufficient
Together: Still Not sufficient

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

ill solve it next time
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Re: Inequality and absolute value questions from my collection [#permalink] New post 16 Aug 2010, 07:10
An excellent resource. Thanks Bunuel
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Re: Inequality and absolute value questions from my collection [#permalink] New post 21 Oct 2010, 21:54
Great stuff.

Learned a lot from this set.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 31 Oct 2010, 23:55
Dear Bunuel,

I don't quite understand your explanation on question 4, could you please explain to me again?.
When I solved both statement 1 and 2, the answer can be + or -, therefore, how the answer comes up with only 1 sign?.

Thanks
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Re: Inequality and absolute value questions from my collection [#permalink] New post 01 Nov 2010, 00:04
sumitra wrote:
Dear Bunuel,

I don't quite understand your explanation on question 4, could you please explain to me again?.
When I solved both statement 1 and 2, the answer can be + or -, therefore, how the answer comes up with only 1 sign?.

Thanks
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Welcome to GMATClub !


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


(1) 2(x-y)=1 OR (x-y)=0.5
Does not need both to be positive or not, EG. x=5, y=4.5 ... x=-4.5, y=-5

(2) x/y>1
Again not sufficient. EG. x=5, y=2 ... x=-5,y=-2

(1+2) x/y>1 does mean that the sign of x & y has to be the same since their ratio is greater than 0
Now (x-y)=0.5 is also true which means x is greater than y
Either x&y are both negative or both positive
If x and y are both negative, and we know x is greater than y, then (x/y) will be less than 1 (x=-3, y=-4 .. x/y=3/4<1) which contradicts statement 1
Hence x & y must both be positive
Sufficient

Answer is (C) : Both statements together but neither alone is sufficient
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Re: Inequality and absolute value questions from my collection [#permalink] New post 01 Nov 2010, 06:46
Expert's post
sumitra wrote:
Dear Bunuel,

I don't quite understand your explanation on question 4, could you please explain to me again?.
When I solved both statement 1 and 2, the answer can be + or -, therefore, how the answer comes up with only 1 sign?.

Thanks
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Hi, and welcome to Gmat Club.

I guess you understand why each statement alone is not sufficient. As for (1)+(2):

From (1): 2x-2y=1 --> x=y+\frac{1}{2}

From (2) \frac{x}{y}>1, we can only deduce that x and y have the same sigh (either both positive or both negative).

When we consider two statement together:

From (2): \frac{x}{y}>1 --> \frac{x}{y}-1>0 --> \frac{x-y}{y}>0 --> substitute x from (1) --> \frac{y+\frac{1}{2}-y}{y}>0--> \frac{1}{2y}>0 (we can drop 2 as it won't affect anything here and write as I wrote \frac{1}{y}>0, but basically it's the same) --> \frac{1}{2y}>0 means y is positive, and from (2) we know that if y is positive x must also be positive.

OR: as y is positive and as from (1) x=y+\frac{1}{2}, x=positive+\frac{1}{2}=positive, hence x is positive too.

Hope it helps.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 11 Dec 2010, 22:35
Hi
for question#12, the answer should be E. Lets take some examples. Let r = -5, s=5. Then both the statements 1 and 2 are valid. But r is not equal to s. when r = 3 and s = 3, then also both the statements are valid, but r = s.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 19 Dec 2010, 02:12
Bunuel wrote:
4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.

(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally.

One of the approaches:
2x-2y=1 --> x=y+\frac{1}{2}
\frac{x}{y}>1 --> \frac{x-y}{y}>0 --> substitute x --> \frac{1}{y}>0 --> y is positive, and as x=y+\frac{1}{2}, x is positive too. Sufficient.

Answer: C.


Hi

I am not sure if the answer is C.From my point of view it needs to be E.Reason is even if we combine both I and II ,the answer is insufficient.

Consider putting the values as 1) x=1 y=1/2 ,now x>y and x-y=1/2.In this case both x and y are positive
Now2) x=-1 and y=-1/2,now again x>y and x-y=-1/2.In this case both are negative.

Hence insufficient to deduce whther x and y are both positive.

Please share your inputs on the same.
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Re: Inequality and absolute value questions from my collection [#permalink] New post 19 Dec 2010, 02:27
Expert's post
Eshika wrote:
Bunuel wrote:
4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

(1) 2x-2y=1. Well this one is clearly insufficient. You can do it with number plugging OR consider the following: x and y both positive means that point (x,y) is in the I quadrant. 2x-2y=1 --> y=x-1/2, we know it's an equation of a line and basically question asks whether this line (all (x,y) points of this line) is only in I quadrant. It's just not possible. Not sufficient.

(2) x/y>1 --> x and y have the same sign. But we don't know whether they are both positive or both negative. Not sufficient.

(1)+(2) Again it can be done with different approaches. You should just find the one which is the less time-consuming and comfortable for you personally.

One of the approaches:
2x-2y=1 --> x=y+\frac{1}{2}
\frac{x}{y}>1 --> \frac{x-y}{y}>0 --> substitute x --> \frac{1}{y}>0 --> y is positive, and as x=y+\frac{1}{2}, x is positive too. Sufficient.

Answer: C.


Hi

I am not sure if the answer is C.From my point of view it needs to be E.Reason is even if we combine both I and II ,the answer is insufficient.

Consider putting the values as 1) x=1 y=1/2 ,now x>y and x-y=1/2.In this case both x and y are positive
Now2) x=-1 and y=-1/2,now again x>y and x-y=-1/2.In this case both are negative.

Hence insufficient to deduce whther x and y are both positive.

Please share your inputs on the same.


OA for this question is C, not E.

The red part in your reasoning is not correct: if x=-1 and y=-1/2 then y>x.

Here is the logic for C:

From (2) \frac{x}{y}>1, we can only deduce that x and y have the same sigh (either both positive or both negative).

When we consider two statement together:

From (1): 2x-2y=1 --> x=y+\frac{1}{2}

From (2): \frac{x}{y}>1 --> \frac{x}{y}-1>0 --> \frac{x-y}{y}>0 --> substitute x from (1) --> \frac{y+\frac{1}{2}-y}{y}>0--> \frac{1}{2y}>0 (we can drop 2 as it won't affect anything here and write as I wrote \frac{1}{y}>0, but basically it's the same) --> \frac{1}{2y}>0 means y is positive, and from (2) we know that if y is positive x must also be positive.

OR: as y is positive and as from (1) x=y+\frac{1}{2}, x=positive+\frac{1}{2}=positive, hence x is positive too.

Does it make sense now?
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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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Re: Inequality and absolute value questions from my collection [#permalink] New post 19 Dec 2010, 02:48
Oh yeah..my mistake again...Thanks again for your prompt reply...I guess more concentration is required from my end..Thanks for your help...
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Re: Inequality and absolute value questions from my collection [#permalink] New post 05 Feb 2011, 08:26
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.
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Re: Inequality and absolute value questions from my collection   [#permalink] 05 Feb 2011, 08:26
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