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Inequality and absolute value questions from my collection [#permalink]
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16 Nov 2009, 11:33
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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: inequalityandabsolutevaluequestionsfrommycollection8693920.html#p6536902. If y is an integer and \(y = x + x\), is \(y = 0\)? (1) \(x < 0\) (2) \(y < 1\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693920.html#p6536953. Is \(x^2 + y^2 > 4a\)?(1) \((x + y)^2 = 9a\) (2) \((x – y)^2 = a\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p6536974. Are x and y both positive?(1) \(2x2y=1\) (2) \(\frac{x}{y}>1\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p6537095. What is the value of y?(1) \(3x^2 4 = y  2\) (2) \(3  y = 11\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p6537316. If x and y are integer, is y > 0? (1) \(x +1 > 0\) (2) \(xy > 0\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p6537407. \(x+2=y+2\) what is the value of x+y?(1) \(xy<0\) (2) \(x>2\), \(y<2\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p653783 AND inequalityandabsolutevaluequestionsfrommycollection86939160.html#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: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p6537899. Is n<0?(1) \(n=n\) (2) \(n^2=16\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p65379210. If n is not equal to 0, is n < 4 ?(1) \(n^2 > 16\) (2) \(\frac{1}{n} > n\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p65379611. Is \(x+y>xy\)?(1) \(x > y\) (2) \(xy < x\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p65385312. Is r=s?(1) \(s \leq r \leq s\) (2) \(r \geq s\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p65387013. Is \(x1 < 1\)?(1) \((x1)^2 \leq 1\) (2) \(x^2  1 > 0\) Solution: inequalityandabsolutevaluequestionsfrommycollection8693940.html#p653886Official answers (OA's) and detailed solutions are in my posts on pages 2 and 3.
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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 DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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16 Nov 2009, 12:42
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ahh..yes...fresh meat

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16 Nov 2009, 12:51
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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 [#permalink]
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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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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 [#permalink]
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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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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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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 [#permalink]
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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 [#permalink]
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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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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 DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
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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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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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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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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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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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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.
Last edited by h2polo on 17 Nov 2009, 10:54, edited 1 time in total.

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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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17 Nov 2009, 10:48
4) I) 2x2y=1 so y=x1/2 NS II)x/y>0 so x and y have the same sign and the modulus of x has to be larger than the modulus of y NS Together, to satisfy both clues needs to be larger than 1/2 and x becomes larger than 0; the stem is true, therefore C

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h2polo wrote: 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 Your last choice of numbers: x=1/2, y=1/2 does not satisfy clue I, because 2*(1/2)2*(1/2)=2, not 1

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