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Inequality and absolute value questions from my collection
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16 Nov 2009, 11:33
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: https://gmatclub.com/forum/inequalitya ... 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#p653709 Graphic approach: https://gmatclub.com/forum/inequalitya ... l#p12698025. 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 https://gmatclub.com/forum/inequalitya ... 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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Re: Inequality and absolute value questions from my collection
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18 Nov 2009, 08:39
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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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 15:29
Quote: 11. Is x+y>xy? (1) x > y (2) xy < x Question Stem : Is the distance between X and Y greater than the distance between X and Y?
Note : Using number line approach. St. (1) : x > y
OR, The distance between X and the origin is greater than the distance between Y and the origin. Now we can have two cases : (a) When X is positive : In this case, X > Y for the above condition to be true. _______________________________________________________ _________________Y______Origin______Y........ Region of X......... Thus we can see that the distance between X and  Y will always be greater than the distance between X and Y. Hence question stem is true. (b) When X is negative : In this case, X < Y for the statement to be true. _______________________________________________________ .... Region of X.....Y_____Origin_______Y___________ Thus we can see that the distance between X and Y will always be less than the distance between X and Y. Hence, question stem becomes false. Due to conflicting statements, St (1) becomes Insufficient. St. (2) : xy < x
OR, the distance between X and Y is less than the distance between X and the origin. Now again let us consider the different cases : (a) When X is positive : For the statement to be true and for X to be positive, X must be greater than Y/2. For any value of X less than Y/2 the statement will become false. The statement will be true for any value greater than Y/2. Thus we see that only one case comes into the picture. Now let us see how it relates to the question stem. ______________________;_____________________ ________Y____Origin__y/2...Y...... Region of X.... Thus we can see that the distance between X and  Y will be greater than the distance between X and Y for all values of X > Y/2. Thus question stem is true. St. (2) is sufficient. Answer : B
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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 12:42
ahh..yes...fresh meat



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 12:51
Bunuel wrote: 2. If y is an integer and y = x + x, is y = 0? (1) x < 0 (2) y < 1
1. x < 0 you will always get x minus itself so always 0 2. y < 1 y is an integer so y<=0 y can't be negative because x minus itself is always zero answer d



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 13:08
Bunuel wrote: 13. Is x1 < 1? (1) (x1)^2 <= 1 (2) x^2  1 > 0
I'm getting B for this one 1. (x1)^2 <= 1 x can be 0 which would make the question no or x can be 1/2 which would make the answer yes so 1 is insufficient 2. x^2  1 > 0 means x^2>1 so x<1 or x>1 both of which make the question no so sufficient



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 13:19
Bunuel wrote:
12. Is r=s? (1) s<=r<=s (2) r>=s
I'm getting c 1. s can be 3 and r can be 3 which makes question yes s can be 3 and r can be 2 which makes question no insufficient 2. r can be 3 and s can be 3 makes question yes r can be 3 s can be 2 makes question no insufficient combining: r>=s means r>=s or r<=s and s<=r<=s means s<=r and r<=s now we have s<=r and s>=r so s = r or s = r r>=s and r<=s so s = r



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 15:33
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



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 20:07
Bunuel, thanks for the questions. Please provide the OA's too. It would be great if you can provide them soon. I am having my GMAT this week, so kinda tensed and impatient. Also, I am yet to give my MGMAT CAT's, so tell me whether should I solve the questions on the forum because if the questions are from the MGMAT CAT's or Gmat Prep then it may overestimate my result. I would appreciate your response. Thanks once again.



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 21:39
Quality questions as always... Thanks Bunuel! +1



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Re: Inequality and absolute value questions from my collection
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16 Nov 2009, 22:46
lagomez wrote: Bunuel wrote: 13. Is x1 < 1? (1) (x1)^2 <= 1 (2) x^2  1 > 0
I'm getting B for this one 1. (x1)^2 <= 1 x can be 0 which would make the question no or x can be 1/2 which would make the answer yes so 1 is insufficient 2. x^2  1 > 0 means x^2>1 so x<1 or x>1 both of which make the question no so sufficient (1) (x1)^2 <= 1 x is 0 to 2. If x = 2, yes. If x < 2, No. (2) x^2  1 > 0 x cannot be 1 to 1 i.e. x<1 or x>1. NSF. From 1 and 2: x is >1 but <=2. NSF.. E.
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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 03:15
gmat620 wrote: Bunuel, thanks for the questions. Please provide the OA's too. It would be great if you can provide them soon. I am having my GMAT this week, so kinda tensed and impatient. Also, I am yet to give my MGMAT CAT's, so tell me whether should I solve the questions on the forum because if the questions are from the MGMAT CAT's or Gmat Prep then it may overestimate my result. I would appreciate your response. Thanks once again. These questions are from various sources. Couple of questions might be from MGMAT CAT or Gmat Prep, but not more than that. I'll provide OA in a day or two, after discussions. Tell me if you want the answers for the specific questions earlier than that and I'll mail you.
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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 05:18
Bunuel 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
Not sure about this one... First I reduced the given equation (divided out the y) and solved for x: 6*x*y = x^2*y + 9*y 6*x = x^2 + 9 0 = x^2  6*x + 9 0 = (x3)^2 x = 3 Statement 1: yx=3 y3=3 y=6 xy=3*6=18 SUFFICIENT Statement 2: x^3<0 We have no idea what the value of y is from this statement. The only thing that made me look twice was the face that if x^3 is true, then x should be a negative value... did I calculate the value of x incorrectly above? INSUFFICIENT ANSWER: A.



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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 05:34
Bunuel wrote: 2. If y is an integer and y = x + x, is y = 0? (1) x < 0 (2) y < 1
Another way of looking at the problem is to ask, is x<0? Because if it is, then we know that y is zero. The only case in which y will not be zero is if x is positive. Statement 1: x<0... answers my question above. SUFFICIENT Statement 2: y<1 Because y is an integer, it must be one of the following values: 0, 1, 2, 3... BUT x + x can never be a negative value. The lowest value that it can be is 0. Hence, y can never be negative and the only possible value it can be then is 0. SUFFICIENT ANSWER: D.



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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 07:38
3) I) (x+y)^2=9a x^2+y^2=9a2xy NS II) (xy)^2=a x^2+y^2=a+2xy NS Together 2(x^2+y^2)=10a x^2+y^2=5a If either x or y are larger than 0, the stem would be true, but if they’re both zero the stem is false, hence E
4) I don’t get the two clues; they seem to be mutually exclusive
5) I) 3x^24=y2 either y=3x^210 or y=143x^2 NS II) 3y=11 either y=8 or y=14 NS Together 8=3x^210 so 3x^2=2 ok 14=3x^210 so 3x^2=28 ok, hence E
6) I) x+1>0 so x={0, 1, 2, …} NS II) xy>0 so x and y have the same sign and none of them is zero NS Together, x={1, 2, 3, ..} and y has the same sign, hence C
7) x+2=y+2 either x+2=y+2 or x+2=y2 (the other two combinations can be transformed into these by multiplying by 1) Reordering: xy=0 or x+y=4 I)xy<0, hence x and y have different signs and none of them is zero. The only possibility is x+y=4 S II) x>2, y<2 hence x#y. The only possibility is x+y=4 S, therefore D
8)a*b#0, hence a and b are both nonzero I) a*b=a*b a and b have the same sign and the stem is always true S II) a/b=a/b this is true regardless of the values of a and b, and nothing can be said about the stem NS, therefore A
9) I) –n=n n<=0 NS II) n^2=16 n=+/4 NS Together n=4 therefore C
10)n#0 I) n^2>16, so n>4 S II) 1/n>n true for n<1 NS, therefore A
11) Plugging in numbers I get B, but there’s no rime or reason to my solution
12) I) –s<=r<=s obviously NS. Since s>=s, s is either positive or zero II)r>=s obviously NS Together: I) tells us that s>=0; II) tells us that r>=s or r<=s. The only case in which I and II are simultaneously satisfied is r=s, therefore C
13) x=(0:2) with 0 and 2 excluded I) (x1)^2<=1, hence x=[0:2] with 0 and 2 included, hence NS II) x^21>0 x<1 or x>1. For x=1.5 the stem is true, for x=3 it is false, hence NS Together, for x=1.5 the stem is true, for x=2 it is false, hence E



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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 10:07
Marco83 wrote: 4) I don’t get the two clues; they seem to be mutually exclusive
Yes there was a typo in 4. Edited. Great job Marco83. Even though not every answer is correct, you definitely know how to deal with this kind of problems.
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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 10:13
4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
Statement 1:
2(1)2(1/2)=1 , x,y are both positve
2(1/2)2(1/2)=1 x is positive, y is negative
INSUFFICIENT
Statement 2:
Either (x,y) are both positive or both negative
INSUFFICENT
Statement 1 and 2:
With both requirements x must be greater than y and satisfy this equation: 2x2y=1
2(1)2(1/2)=1 , x,y are both positve and x>y
2(1/2)2(1/2)=1 x is positive, y is negative and x>y
Answer: E



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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 10:27
12. Is r=s? (1) s<=r<=s (2) r>=s E – for this  both can be true or false when 0< r < 1 For example , take r as 0.8 S = 0.86 i.e. 0.86 < = 0.8 < = 0.86 0.8>= 0.86 i.e. 1 >= 0.86 Combining , any values can be taken , on values > =1 , both r and s will be same 3. Is x^2 + y^2 > 4a? (1) (x + y)^2 = 9a (2) (x – y)^2 = a C is the answer Combined both and the equation will give x^2 + y^2 = 5a
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Re: Inequality and absolute value questions from my collection
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Updated on: 17 Nov 2009, 10:54
Bunuel wrote: 5. What is the value of y? (1) 3x^2 4 = y  2 (2) 3  y = 11
Statement 1: Two equations, two unknowns... INSUFFICIENT Statement 2: 3  y = 11 (3y)=11 or (3y)=11 y=8, 14 INSUFFICIENT Statements 1 and 2: y must be 14 because 3x^2 4 can never be a negative value (no matter what you plug in for x, you will get a positve value because of the absolute value signs). SUFFICIENT ANSWER: C.
Originally posted by h2polo on 17 Nov 2009, 10:34.
Last edited by h2polo on 17 Nov 2009, 10:54, edited 1 time in total.



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Re: Inequality and absolute value questions from my collection
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17 Nov 2009, 10:43
Bunuel wrote: 6. If x and y are integer, is y > 0? (1) x +1 > 0 (2) xy > 0
Statement 1: Nothing about y... INSUFFICIENT Statement 2: two equations, two unknowns... INSUFFICIENT Statements 1 and 2: From x +1 > 0 and the fact that x must be an integer, we know that x must be [0,1,2,3...] Because we know that xy > 0, we know that x cannot be 0... therefore y must be a positive integer! SUFFICIENT ANSWER: C.




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