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Inequality and absolute value questions from my collection [#permalink]
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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: http://gmatclub.com/forum/inequalityan ... ml#p6536902. If y is an integer and \(y = x + x\), is \(y = 0\)? (1) \(x < 0\) (2) \(y < 1\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6536953. Is \(x^2 + y^2 > 4a\)?(1) \((x + y)^2 = 9a\) (2) \((x – y)^2 = a\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6536974. Are x and y both positive?(1) \(2x2y=1\) (2) \(\frac{x}{y}>1\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6537095. What is the value of y?(1) \(3x^2 4 = y  2\) (2) \(3  y = 11\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6537316. If x and y are integer, is y > 0? (1) \(x +1 > 0\) (2) \(xy > 0\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6537407. \(x+2=y+2\) what is the value of x+y?(1) \(xy<0\) (2) \(x>2\), \(y<2\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p653783 AND http://gmatclub.com/forum/inequalityan ... l#p11117478. \(a*b \neq 0\). Is \(\frac{a}{b}=\frac{a}{b}\)?(1) \(a*b=a*b\) (2) \(\frac{a}{b}=\frac{a}{b}\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p6537899. Is n<0?(1) \(n=n\) (2) \(n^2=16\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p65379210. If n is not equal to 0, is n < 4 ?(1) \(n^2 > 16\) (2) \(\frac{1}{n} > n\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p65379611. Is \(x+y>xy\)?(1) \(x > y\) (2) \(xy < x\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p65385312. Is r=s?(1) \(s \leq r \leq s\) (2) \(r \geq s\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p65387013. Is \(x1 < 1\)?(1) \((x1)^2 \leq 1\) (2) \(x^2  1 > 0\) Solution: http://gmatclub.com/forum/inequalityan ... ml#p653886Official answers (OA's) and detailed solutions are in my posts on pages 2 and 3.PLEASE READ THE WHOLE DISCUSSION BEFORE POSTING A QUESTION.
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21 Jan 2016, 21:27
Bunuel wrote: 13. Is x1 < 1? (1) (x1)^2 <= 1 (2) x^2  1 > 0
Last one.
Is x1 < 1? Basically the question asks is 0<x<2 true?
(1) (x1)^2 <= 1 > x^22x<=0 > x(x2)<=0 > 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.
(2) x^2  1 > 0 > x<1 or x>1. Not sufficient.
(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.
Answer: E. Hi: I have been trying to comprehend the concepts on this topic lately. Please could you help me understand where am i going fundamentally wrong with the below approach, not particularly in the context of this question though: Option 2: x^2  1>0 > x^2  1^2>0 > (x+1)(x1)>0 > X>1; x>1



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Re: Inequality and absolute value questions from my collection [#permalink]
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24 Jan 2016, 07:19
7. x+2=y+2 what is the value of x+y?[/b] (1) xy<0 (2) x>2 y<2 (1) xy<0 > We have scenario B, hence x+y=4. Sufficient
bunuel...
if X+Y=4 then x=Y=2(as derived) but xy=4(as per condition XY<0) should it be as X+Y=2
Plz clear me i m confused



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24 Jan 2016, 07:26
Bunuel wrote: 7. x+2=y+2 what is the value of x+y? (1) xy<0 (2) x>2 y<2 This one is quite interesting.
First note that x+2=y+2 can take only two possible forms:
A. x+2=y+2 > x=y. This will occur if and only x and y are both >= than 2 OR both <= than 2. In that case x=y. Which means that their product will always be positive or zero when x=y=0. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=4. Also note that viseversa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.
(1) xy<0 > We have scenario B, hence x+y=4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=4. Sufficient.
Answer: D. (1) xy<0 > We have scenario B, hence x+y=4. Sufficient bunuel... if X+Y=4 then x=Y=2(as derived) but xy=4(as per condition XY<0) should it be as X+Y=2 Plz clear me i m confused



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Re: Inequality and absolute value questions from my collection [#permalink]
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21 Feb 2016, 22:37
4. Are x and y both positive? (1) 2x−2y=1 (2) x/y>1
after solving for statement 1 and 2 we are left with option C & E as A & B are NS individually
now, i get XY=1/2 X/Y>1 (by this statement we know that X & Y are both either Positive or Negative and x>y) now, we put values to check whether Statement 1 is holding true for what Statement 2 implies. Both Positive : x=4 and y=3.5 we get 1/2 Both negative : x=4 and y=3.5 we get 1/2 which is not correct so, C= sufficient



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Re: Inequality and absolute value questions from my collection [#permalink]
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23 Feb 2016, 00:25
Bunuel wrote: 9. Is n<0? (1) n=n (2) n^2=16
(1) n=n, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.
(2) n^2=16 > n=4 or n=4. Not sufficient.
(1)+(2) n is negative OR n equals to zero from (1), n is 4 or 4 from (2). > n=4, hence it's negative, sufficient.
Answer: C. hey bunuel can you please clear my doubt? in statement 1 you've written either n is negative OR n equals to zero but as per my knowledge shouldn't n be negative only because I've read it in many post that are on absolute value, here's a link: mathabsolutevaluemodulus86462.htmlcorrect me if i'm wrong!



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23 Feb 2016, 00:43
nishantdoshi wrote: Bunuel wrote: 9. Is n<0? (1) n=n (2) n^2=16
(1) n=n, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.
(2) n^2=16 > n=4 or n=4. Not sufficient.
(1)+(2) n is negative OR n equals to zero from (1), n is 4 or 4 from (2). > n=4, hence it's negative, sufficient.
Answer: C. hey bunuel can you please clear my doubt? in statement 1 you've written either n is negative OR n equals to zero but as per my knowledge shouldn't n be negative only because I've read it in many post that are on absolute value, here's a link: mathabsolutevaluemodulus86462.htmlcorrect me if i'm wrong! An absolute value cannot be negative but it CAN be 0. For this particular case 0 fits: n=n > 0 = 0 > 0 = 0.
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23 Feb 2016, 00:53
Bunuel wrote: nishantdoshi wrote: Bunuel wrote: 9. Is n<0? (1) n=n (2) n^2=16
(1) n=n, means that either n is negative OR n equals to zero. We are asked whether n is negative so we can not be sure. Not sufficient.
(2) n^2=16 > n=4 or n=4. Not sufficient.
(1)+(2) n is negative OR n equals to zero from (1), n is 4 or 4 from (2). > n=4, hence it's negative, sufficient.
Answer: C. hey bunuel can you please clear my doubt? in statement 1 you've written either n is negative OR n equals to zero but as per my knowledge shouldn't n be negative only because I've read it in many post that are on absolute value, here's a link: mathabsolutevaluemodulus86462.htmlcorrect me if i'm wrong! An absolute value cannot be negative but it CAN be 0. For this particular case 0 fits: n=n > 0 = 0 > 0 = 0. thanks for the reply my understanding about this topic is that... if x>=0 then x=xand if x<0 then x=x am i wrong? please reply!!!



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23 Feb 2016, 01:06
Bunuel wrote: 7. x+2=y+2 what is the value of x+y? (1) xy<0 (2) x>2 y<2 This one is quite interesting.
First note that x+2=y+2 can take only two possible forms:
A. x+2=y+2 > x=y. This will occur if and only x and y are both >= than 2 OR both <= than 2. In that case x=y. Which means that their product will always be positive or zero when x=y=0. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=4. Also note that viseversa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.
(1) xy<0 > We have scenario B, hence x+y=4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=4. Sufficient.
Answer: D. so it should be if x>=0 then x=xif x<=0 then x=xcorrect? thanks in advance



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23 Feb 2016, 01:15
nishantdoshi wrote: Bunuel wrote: 7. x+2=y+2 what is the value of x+y? (1) xy<0 (2) x>2 y<2 This one is quite interesting.
First note that x+2=y+2 can take only two possible forms:
A. x+2=y+2 > x=y. This will occur if and only x and y are both >= than 2 OR both <= than 2. In that case x=y. Which means that their product will always be positive or zero when x=y=0. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=4. Also note that viseversa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.
(1) xy<0 > We have scenario B, hence x+y=4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=4. Sufficient.
Answer: D. so it should be if x>=0 then x=xif x<=0 then x=xcorrect? thanks in advance __________________________ Yes.
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13 Apr 2016, 10:58
Hi Bunuel,
(1) (x1)^2 <= 1 > x^22x<=0 > x(x2)<=0 > 0<=x<=2. x is in the range (0,2) inclusive. I have some difficulties to understand this. you clarified this above but I did not get it. please make simple for me please. algebraic steps lead to 2 <= X <=0



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14 May 2016, 00:34
In your step you have taken it as (x3)^2 but isn't (x^26x+9) = (x3)^2 or (3x)^2? Also, what does 'reduce the expression by y' mean? Bunuel wrote: SOLUTIONS:
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 let's simplify given expression \(6*x*y = x^2*y + 9*y\):
\(y*(x^26x+9)=0\) > \(y*(x3)^2=0\). Note here that we CAN NOT reduce this expression by \(y\), as some of you did. Remember we are asked to determine the value of \(xy\), and when reducing by \(y\) you are assuming that \(y\) doesn't equal to \(0\). We don't know that.
Next: we can conclude that either \(x=3\) or/and \(y=0\). Which means that \(xy\) equals to 0, when y=0 and x any value (including 3), OR \(xy=3*y\) when y is not equal to zero, and x=3.
(1) \(yx=3\). If y is not 0, x must be 3 and yx to be 3, y must be 6. In this case \(xy=18\). But if y=0 then x=3 and \(xy=0\). Two possible scenarios. Not sufficient.
OR:
\(yx=3\) > \(x=y3\) > \(y*(x3)^2=y*(y33)^2=y(y6)^2=0\) > either \(y=0\) or \(y=6\) > if \(y=0\), then \(x=3\) and \(xy=0\) \(or\) if \(y=6\), then \(x=3\) and \(xy=18\). Two different answers. Not sufficient.
(2) \(x^3<0\). x is negative, hence x is not equals to 3, hence y must be 0. So, xy=0. Sufficient.
Answer: B.
This one was quite tricky and was solved incorrectly by all of you.
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.
Never multiply (or reduce) inequality by variable (or expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero.



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14 May 2016, 01:06
ameyaprabhu wrote: In your step you have taken it as (x3)^2 but isn't (x^26x+9) = (x3)^2 or (3x)^2? Also, what does 'reduce the expression by y' mean? Bunuel wrote: SOLUTIONS:
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 let's simplify given expression \(6*x*y = x^2*y + 9*y\):
\(y*(x^26x+9)=0\) > \(y*(x3)^2=0\). Note here that we CAN NOT reduce this expression by \(y\), as some of you did. Remember we are asked to determine the value of \(xy\), and when reducing by \(y\) you are assuming that \(y\) doesn't equal to \(0\). We don't know that.
Next: we can conclude that either \(x=3\) or/and \(y=0\). Which means that \(xy\) equals to 0, when y=0 and x any value (including 3), OR \(xy=3*y\) when y is not equal to zero, and x=3.
(1) \(yx=3\). If y is not 0, x must be 3 and yx to be 3, y must be 6. In this case \(xy=18\). But if y=0 then x=3 and \(xy=0\). Two possible scenarios. Not sufficient.
OR:
\(yx=3\) > \(x=y3\) > \(y*(x3)^2=y*(y33)^2=y(y6)^2=0\) > either \(y=0\) or \(y=6\) > if \(y=0\), then \(x=3\) and \(xy=0\) \(or\) if \(y=6\), then \(x=3\) and \(xy=18\). Two different answers. Not sufficient.
(2) \(x^3<0\). x is negative, hence x is not equals to 3, hence y must be 0. So, xy=0. Sufficient.
Answer: B.
This one was quite tricky and was solved incorrectly by all of you.
Never reduce equation by variable (or expression with variable), if you are not certain that variable (or expression with variable) doesn't equal to zero. We can not divide by zero.
Never multiply (or reduce) inequality by variable (or expression with variable) if you don't know the sign of it or are not certain that variable (or expression with variable) doesn't equal to zero. Both (x3)^2 and (3x)^2 are the same.
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08 Jun 2016, 00:40
Bunuel wrote: 7. x+2=y+2 what is the value of x+y? (1) xy<0 (2) x>2 y<2 This one is quite interesting.
First note that x+2=y+2 can take only two possible forms:
A. x+2=y+2 > x=y. This will occur if and only x and y are both >= than 2 OR both <= than 2. In that case x=y. Which means that their product will always be positive or zero when x=y=0. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=4. Also note that viseversa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.
(1) xy<0 > We have scenario B, hence x+y=4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=4. Sufficient.
Answer: D. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2. why we did not consider the case when 2+(2)= 4, when x= 2 and y= 2. Also clarify when we get 2 scenarios from the question we have to satisfy either of them from given statements.



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11 Jun 2016, 09:43
Bunuel wrote: 7. x+2=y+2 what is the value of x+y? (1) xy<0 (2) x>2 y<2 This one is quite interesting.
First note that x+2=y+2 can take only two possible forms:
A. x+2=y+2 > x=y. This will occur if and only x and y are both >= than 2 OR both <= than 2. In that case x=y. Which means that their product will always be positive or zero when x=y=0. B. x+2=y2 > x+y=4. This will occur when either x or y is less then 2 and the other is more than 2.
When we have scenario A, xy will be nonnegative only. Hence if xy is negative we have scenario B and x+y=4. Also note that viseversa is not right. Meaning that we can have scenario B and xy may be positive as well as negative.
(1) xy<0 > We have scenario B, hence x+y=4. Sufficient.
(2) x>2 and y<2, x is not equal to y, we don't have scenario A, hence we have scenario B, hence x+y=4. Sufficient.
Answer: D. Why didnt u consider x+2 = y2 => x+y =0 Then for xy<0 we get two values of x+y i.e. 0 and 4



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11 Jun 2016, 10:12
Bunuel wrote: 10. If n is not equal to 0, is n < 4 ? (1) n^2 > 16 (2) 1/n > n
Question basically asks is 4<n<4 true.
(1) n^2>16 > n>4 or n<4, the answer to the question is NO. Sufficient.
(2) 1/n > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.
Answer: A. How is stmt 1 sufficient? n^2>16 => n<4 U n>4 Now if n = 5 => n is not less than 4 If n = 3 => n is less than 4 We are getting two different answers.



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11 Jun 2016, 18:00
sa18 wrote: Bunuel wrote: 10. If n is not equal to 0, is n < 4 ? (1) n^2 > 16 (2) 1/n > n
Question basically asks is 4<n<4 true.
(1) n^2>16 > n>4 or n<4, the answer to the question is NO. Sufficient.
(2) 1/n > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.
Answer: A. How is stmt 1 sufficient? n^2>16 => n<4 U n>4 Now if n = 5 => n is not less than 4 If n = 3 => n is less than 4 We are getting two different answers. If n=5 >n =5 = 5 >4. All values provided by s1 will be >4  thus this statement is suff.



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02 Sep 2016, 23:52
Bunuel wrote: 13. Is x1 < 1? (1) (x1)^2 <= 1 (2) x^2  1 > 0
Last one.
Is x1 < 1? Basically the question asks is 0<x<2 true?
(1) (x1)^2 <= 1 > x^22x<=0 > x(x2)<=0 > 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.
(2) x^2  1 > 0 > x<1 or x>1. Not sufficient.
(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.
Answer: E. Hi Bunuel, Can you please explain why have you not considered (for option1) the other case. I mean, x(x2)<=0 can lead to two possiblities, one that you have mentioned, and the other one could be just the opposite x<=0 and x>=2. I need to understand this, please reply. Thanks.



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Re: Inequality and absolute value questions from my collection [#permalink]
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04 Sep 2016, 04:50
rpradhan25 wrote: Bunuel wrote: 13. Is x1 < 1? (1) (x1)^2 <= 1 (2) x^2  1 > 0
Last one.
Is x1 < 1? Basically the question asks is 0<x<2 true?
(1) (x1)^2 <= 1 > x^22x<=0 > x(x2)<=0 > 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.
(2) x^2  1 > 0 > x<1 or x>1. Not sufficient.
(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.
Answer: E. Hi Bunuel, Can you please explain why have you not considered (for option1) the other case. I mean, x(x2)<=0 can lead to two possiblities, one that you have mentioned, and the other one could be just the opposite x<=0 and x>=2. I need to understand this, please reply. Thanks. x(x2)<=0 is true for 0<=x<=2 and not true for any other range. Check the links below: Inequalities Made Easy!Solving Quadratic Inequalities  Graphic ApproachInequality tipsWavy Line Method Application  Complex Algebraic InequalitiesDS Inequalities Problems PS Inequalities Problems 700+ Inequalities problemsinequalitiestrick91482.htmldatasuffinequalities109078.htmlrangeforvariablexinagiveninequality109468.htmleverythingislessthanzero108884.htmlgraphicapproachtoproblemswithinequalities68037.htmlHope it helps.
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Re: Inequality and absolute value questions from my collection
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