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Re: Inequality and absolute value questions from my collection
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01 Jun 2015, 00:59
rohitd80 wrote: lagomez wrote: 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
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.
(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. would you think this is a 700+ question? Hi Bunuel, Can you please comment if my approach is correct? Given: 6*x*y = x^2*y + 9*y, which is reduced to y*(x3)^2 = 0.......(1) Statement [1] yx=3 or y = x+3 sub'ing y in (1) (x+3)(x3)(x3) = 0 (x^29)(x3)=0 so, x^2 = 9 or x = +/3 and x=3 therefore, y = 6 or 0 for x = 3 and 3 respectively..........Not sufficient Statement [2]....says x < 0, so (x3)^2 will never be = 0, y has to be it.....therefore xy=0 so sufficient! So my question to you is whether I handled [1] properly? and whether substituting y into the original problem equation and arriving at x = +/ 3 is correct? So far, I've learnt that Sqrt(x) will only have +x as solution as GMAT doesn;t consider ve solution, but x^2 =25 will have +/5 as solution so we should consider both solutions for x^2 = 25 in our analysis.....Did I get this right? Please opine. Thanks! Yes, everything tis correct. When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root. That is: \(\sqrt{9} = 3\), NOT +3 or 3; \(\sqrt[4]{16} = 2\), NOT +2 or 2; Notice that in contrast, the equation x^2 = 9 has TWO solutions, +3 and 3.
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Re: Inequality and absolute value questions from my collection
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02 Jun 2015, 02:20
camlan1990 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^22xy+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. Hi Bruel, For (1): Because (x + y)^2 <= (1^2+1^2)(x^2+y^2) => x^2+y^2 >= 4.5a > 4a. So A is sufficient? Could you help me find out whether there is any mistake in my solution? Thanks Bruel, Dear camlan1990The highlighted part in your solution above is wrong. The correct expansion for \((x+y)^2 = x^2 + y^2 + 2xy\). You on the other hand have wrongly written: \((x+y)^2 = 2(x^2 + y^2)\). Hope this helped. Best Regards Japinder
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Re: Inequality and absolute value questions from my collection
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02 Jun 2015, 02:30
EgmatQuantExpert wrote: camlan1990 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^22xy+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. Hi Bruel, For (1): Because (x + y)^2 <= (1^2+1^2)(x^2+y^2) => x^2+y^2 >= 4.5a > 4a. So A is sufficient? Could you help me find out whether there is any mistake in my solution? Thanks Bruel, Dear camlan1990The highlighted part in your solution above is wrong. The correct expansion for \((x+y)^2 = x^2 + y^2 + 2xy\). You on the other hand have wrongly written: \((x+y)^2 = 2(x^2 + y^2)\). Hope this helped. Best Regards Japinder Dear Japinder, As I highlighted, (x+y)^2 is smaller or equal 2(x^2 + y^2)



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Re: Inequality and absolute value questions from my collection
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02 Jun 2015, 03:14
camlan1990 wrote: EgmatQuantExpert wrote: camlan1990 wrote: Hi Bruel,
For (1): Because (x + y)^2 <= (1^2+1^2)(x^2+y^2) => x^2+y^2 >= 4.5a > 4a. So A is sufficient?
Could you help me find out whether there is any mistake in my solution? Thanks Bruel, Dear camlan1990The highlighted part in your solution above is wrong. The correct expansion for \((x+y)^2 = x^2 + y^2 + 2xy\). You on the other hand have wrongly written: \((x+y)^2 = 2(x^2 + y^2)\). Hope this helped. Best Regards Japinder Dear Japinder, As I highlighted, (x+y)^2 is smaller or equal 2(x^2 + y^2) Oops camlan1990, that was my bad! I didn't notice the '<' sign in the "<=" Now I do see how you got to x^2+y^2 >= 4.5a But please note that x = 0, y = 0 and a = 0 is one set of values that satisfies this inequality. For these values of x, y and a, the answer to the question 'Is x^2 + y^2 > 4a' is NO For all other values of x, y and a, the answer will be YES. Since we are not able to rule out x, y, a = 0, we cannot infer a unique answer to the posed question using St. 1 alone. So, St. 1 is not sufficient. Hope this helped. Japinder
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Re: Inequality and absolute value questions from my collection
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03 Jun 2015, 02:13
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^22xy+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. Hi Bunuel,
When you combine both statements 1 and 2, why do we need to substitute any value? This is the only part which I didn't understand. Could you please explain why '0' has to be substituted as the question just asks whether x^2 + y^2 = 4a. Nothing is mentioned about what 'a' is.
Awaiting your response.
Thanks.



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Re: Inequality and absolute value questions from my collection
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03 Jun 2015, 02:17
adityanukala 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^22xy+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. Hi Bunuel,
When you combine both statements 1 and 2, why do we need to substitute any value? This is the only part which I didn't understand. Could you please explain why '0' has to be substituted as the question just asks whether x^2 + y^2 = 4a. Nothing is mentioned about what 'a' is.
Awaiting your response.
Thanks.The question asks whether x^2 + y^2 > 4a. If x = y = a = 0, then the answer is NO but if this is not so then the answer is YES. Please read the whole thread. This was discussed many, many, many times before.
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Re: Inequality and absolute value questions from my collection
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03 Jun 2015, 06:14
Bunuel wrote: 4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
(1) 2x2y=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. 2x2y=1 > y=x1/2, 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 timeconsuming and comfortable for you personally.
One of the approaches: \(2x2y=1\) > \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) > \(\frac{xy}{y}>0\) > substitute x > \(\frac{1}{y}>0\) > \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.
Answer: C. Hi Bunuel , I tried substituting values for x and y and seem to be getting a different ans. If we take x=1 and y=1.5 the ans to the question is NO, whereas if we take x=1.5 & y=1 the ans is YES. Since we are unable to ans the question, shouldn't the ans be E? if my reasoning is flawed can you please point out the flaw.. Thanks in advance



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Re: Inequality and absolute value questions from my collection
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03 Jun 2015, 06:34
NavenRk wrote: Bunuel wrote: 4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
(1) 2x2y=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. 2x2y=1 > y=x1/2, 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 timeconsuming and comfortable for you personally.
One of the approaches: \(2x2y=1\) > \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) > \(\frac{xy}{y}>0\) > substitute x > \(\frac{1}{y}>0\) > \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.
Answer: C. Hi Bunuel , I tried substituting values for x and y and seem to be getting a different ans. If we take x=1 and y=1.5 the ans to the question is NO, whereas if we take x=1.5 & y=1 the ans is YES. Since we are unable to ans the question, shouldn't the ans be E? if my reasoning is flawed can you please point out the flaw.. Thanks in advance x = 1 and y = 1.5 does not satisfy x/y > 1.
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Re: Inequality and absolute value questions from my collection
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03 Jun 2015, 06:37
NavenRk wrote: Bunuel wrote: 4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
(1) 2x2y=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. 2x2y=1 > y=x1/2, 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 timeconsuming and comfortable for you personally.
One of the approaches: \(2x2y=1\) > \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) > \(\frac{xy}{y}>0\) > substitute x > \(\frac{1}{y}>0\) > \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.
Answer: C. Hi Bunuel , I tried substituting values for x and y and seem to be getting a different ans. If we take x=1 and y=1.5 the ans to the question is NO, whereas if we take x=1.5 & y=1 the ans is YES. Since we are unable to ans the question, shouldn't the ans be E? if my reasoning is flawed can you please point out the flaw.. Thanks in advance Hi NavinRK, The Flaw is You can NOT take Values x=1 and y=1.5 after combining the two statements because in that case second statement x/y>1 will NOT be satisfied.
Hence the only set of values that all allowed to be taken are the one in which Absolute value of x is greater than Absolute value of y
and
the Sign of both x and y must be same
so only Positive values are acceptable nowI hope it answers your query.
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Re: Inequality and absolute value questions from my collection
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05 Jun 2015, 18:38
Thanks for such post Bunuel,
Pls explain for Q #10 part B) how N will have only negative value when n<1/lnl
Thanks



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Re: Inequality and absolute value questions from my collection
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05 Jun 2015, 21:45
lipsi18 wrote: Thanks for such post Bunuel,
Pls explain for Q #10 part B) how N will have only negative value when n<1/lnl
Thanks Hi Lipsi18, The Explanations is as mentioned below: 10. If n is not equal to 0, is n < 4 ? (1) n^2 > 16 (2) 1/n > nStatement 2: 1/n > nMultiplying n which is a positive value on both sides of the inequation, we get, 1> n*n i.e. n*n < 1 Case 1: n is positive: i.e. n^2 <1 i.e. 0<n<1 Case 2: n is negative n*n <1 is true for all negative values of n But in context of question this statement is NOT SUFFICIENT as it doesn't provide us any specific value of n and for n=5, n >4 and for n=3, n <4 INCONSISTENT answer Hence, NOT SUFFICIENTI hope it answers your query!!!
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Re: Inequality and absolute value questions from my collection
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25 Aug 2015, 02:59
Bunuel wrote: 4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
(1) 2x2y=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. 2x2y=1 > y=x1/2, 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 timeconsuming and comfortable for you personally.
One of the approaches: \(2x2y=1\) > \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) > \(\frac{xy}{y}>0\) > substitute x > \(\frac{1}{y}>0\) > \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.
Answer: C. Also  Viewing graphically A. 2x  2y = 1 Put x = 1, you get y = 1. Put x = 2, you get y = 1.5. Make a rough plot. This is a line which agrees to negative values of both x and y as well. Insufficient. B. x/y > 1 Both x an y can be negative. Insufficient A + B which means for all values x > y, which 2x  2y = 1. This is possible only in the first quadrant whenever the value of x > 1. Sufficient. Answer C sorry for the flipped image
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Re: Inequality and absolute value questions from my collection
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25 Aug 2015, 18:55
Bunuel wrote: 4. Are x and y both positive? (1) 2x2y=1 (2) x/y>1
(1) 2x2y=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. 2x2y=1 > y=x1/2, 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 timeconsuming and comfortable for you personally.
One of the approaches: \(2x2y=1\) > \(x=y+\frac{1}{2}\) \(\frac{x}{y}>1\) > \(\frac{xy}{y}>0\) > substitute x > \(\frac{1}{y}>0\) > \(y\) is positive, and as \(x=y+\frac{1}{2}\), \(x\) is positive too. Sufficient.
Answer: C. I found it easy to rule out (1) and (2) as individually being insufficient. but the conclusion I drew from (2) was obviously that the absolute value of \(x\) has to be bigger than \(y\) (and of course that they are the same size), so regardless of the sign \(2x\) had to be of greater magnitude than \(2y\), and the only way for \(2x2y=1\) was if they were both positive... I know this isn't ground breaking but it's the very simple way I arrived at the correct answer without getting too "mathsy"
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Re: Inequality and absolute value questions from my collection
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27 Aug 2015, 17:39
Bunuel wrote: 12. Is r=s? (1) s<=r<=s (2) r>=s
This one is tough.
(1) s<=r<=s, we can conclude two things from this statement: A. s is either positive or zero, as s<=s; B. r is in the range (s,s) inclusive, meaning that r can be s as well as s. But we don't know whether r=s or not. Not sufficient.
(2) r>=s, clearly insufficient.
(1)+(2) s<=r<=s, s is not negative, r>=s > r>=s or r<=s. This doesn't imply that r=s, from this r can be s as well. Consider: s=5, r=5 > 5<=5<=5 5>=5 s=5, r=5 > 5<=5<=5 5>=5 Both statements are true with these values. Hence insufficient.
Answer: E. I thought of this one graphically... ____________________ s____________ s____________________________ (1) xxxxxxxxxxxxxxxxxrrrrrrrrrrrrrrrrrrrrxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx(2) rrrrrrrrrrrrrrrrrrrrrrrrrxxxxxxxxxxxxxrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr(1)&(2) xxxxxxxxxxxxxrxxxxxxxxxxxxxrxxxxxxxxxxxxxxxxx(1) Everything between and including \(s\) and \(s\) INSUFF(2) Everything outside of but still including \(s\) and \(s\) INSUFF(1) & (2) \(r = s\) or \(s\)... still INSUFF
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Re: Inequality and absolute value questions from my collection
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28 Aug 2015, 11:55
Bunuel wrote: 12. Is r=s? (1) s<=r<=s (2) r>=s
This one is tough.
(1) s<=r<=s, we can conclude two things from this statement: A. s is either positive or zero, as s<=s; B. r is in the range (s,s) inclusive, meaning that r can be s as well as s. But we don't know whether r=s or not. Not sufficient.
(2) r>=s, clearly insufficient.
(1)+(2) s<=r<=s, s is not negative, r>=s > r>=s or r<=s. This doesn't imply that r=s, from this r can be s as well. Consider: s=5, r=5 > 5<=5<=5 5>=5 s=5, r=5 > 5<=5<=5 5>=5 Both statements are true with these values. Hence insufficient.
Answer: E. Can we eliminate B on the basis that it is just a reworded form of Statement A. My understanding is x <1 means 1<x<1. Similarly, s>=r>=s, which is not possible.
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Re: Inequality and absolute value questions from my collection
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28 Aug 2015, 12:29
SauravPathak27 wrote: Bunuel wrote: 12. Is r=s? (1) s<=r<=s (2) r>=s
This one is tough.
(1) s<=r<=s, we can conclude two things from this statement: A. s is either positive or zero, as s<=s; B. r is in the range (s,s) inclusive, meaning that r can be s as well as s. But we don't know whether r=s or not. Not sufficient.
(2) r>=s, clearly insufficient.
(1)+(2) s<=r<=s, s is not negative, r>=s > r>=s or r<=s. This doesn't imply that r=s, from this r can be s as well. Consider: s=5, r=5 > 5<=5<=5 5>=5 s=5, r=5 > 5<=5<=5 5>=5 Both statements are true with these values. Hence insufficient.
Answer: E. Can we eliminate B on the basis that it is just a reworded form of Statement A. My understanding is x <1 means 1<x<1. Similarly, s>=r>=s, which is not possible. Hi SauravPathak27 I'm no expert, but hope I might be able to help. your understanding that x <1 means 1<x<1 is correct. (1) is telling us that \(r\) falls between \(s\) and \(s\) INCLUSIVE of \(s\) and \(s\)...... INSUFFICIENT (2) is telling us that \(r\) falls outside \(s\) and \(s\) INCLUSIVE of \(s\) and \(s\)........ INSUFFICIENT (1) & (2) together tells us that \(r\) must be equal to either \(s\) or \(s\) but cannot determine which one........ INSUFFICIENT Hopefully my freehand below makes it a little clearer. If it helps.... Throw me a Kudos
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Re: Inequality and absolute value questions from my collection
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04 Oct 2015, 11:37
3. Is x^2 + y^2 > 4a? (1) (x + y)^2 = 9a (2) (x – y)^2 = a
(1) (x + y)^2 = 9a x^2 + y^2 + 2xy = 9a x^2 + y^2 = 9a  2xy
(2) (x – y)^2 = a x^2 + y^2  2xy = a x^2 + y^2 = a +2xy
So we can conclude that: 9a  2xy = a + 2xy 8a = 4xy 4a = 2xy
Hence, in the first conclusion is: x^2 + y^2 = 9a  4a x^2 + y^2 = 5a
Answer is C (Together, they can answer the initial question).
Am I wrong, or right?



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Re: Inequality and absolute value questions from my collection
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04 Oct 2015, 12:34
RafaelPina 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 + y^2 + 2xy = 9a x^2 + y^2 = 9a  2xy
(2) (x – y)^2 = a x^2 + y^2  2xy = a x^2 + y^2 = a +2xy
So we can conclude that: 9a  2xy = a + 2xy 8a = 4xy 4a = 2xy
Hence, in the first conclusion is: x^2 + y^2 = 9a  4a x^2 + y^2 = 5a
Answer is C (Together, they can answer the initial question).
Am I wrong, or right? You are correct till \(x^2+y^2=5a\) but what if x=y=0 giving you a=0. In this case, \(x^2+y^2\) will be = 4a and NOT > 4a. This is the reason why E is the correct answer. If you were given "is \(x^2+y^2 \geq 4a\) instead of just >4a", then yes, you would have marked C as the correct answer but you are asked >4a which may or may not be true. Hope this helps.



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Re: Inequality and absolute value questions from my collection
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05 Oct 2015, 21:22
Bunuel wrote: 2. If y is an integer and y = x + x, is y = 0? (1) x < 0 (2) y < 1
Note: as \(y=x+x\) then \(y\) is never negative. For \(x>{0}\) then \(y=x+x=2x\) and for \(x\leq{0}\) then (when x is negative or zero) then \(y=x+x=0\).
(1) \(x<0\) > \(y=x+x=x+x=0\). Sufficient.
(2) \(y<1\), as we concluded y is never negative, and we are given that \(y\) is an integer, hence \(y=0\). Sufficient.
Answer: D. Hi Bunuel, I do agree with option 1 but donot agree with option 2, as the x value can be positive value such as 0.25 etc.so the Y value shall be 0.50, in this case how do we deduce.
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Re: Inequality and absolute value questions from my collection
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06 Oct 2015, 04:57
Sapient wrote: Bunuel wrote: 2. If y is an integer and y = x + x, is y = 0? (1) x < 0 (2) y < 1
Note: as \(y=x+x\) then \(y\) is never negative. For \(x>{0}\) then \(y=x+x=2x\) and for \(x\leq{0}\) then (when x is negative or zero) then \(y=x+x=0\).
(1) \(x<0\) > \(y=x+x=x+x=0\). Sufficient.
(2) \(y<1\), as we concluded y is never negative, and we are given that \(y\) is an integer, hence \(y=0\). Sufficient.
Answer: D. Hi Bunuel, I do agree with option 1 but donot agree with option 2, as the x value can be positive value such as 0.25 etc.so the Y value shall be 0.50, in this case how do we deduce. Statement 2: \(y<1\)Since we know that \(y = x + x\) case 1: x>0.... In this case y = 2x and will be positive case 2: x <0.... In this case y = 0 i.e. Y can never be NegativeThis statement tells us that Y is an Integer less than 1 therefore 0 is the only possible value of y Hence, SUFFICIENT I hope this helps!
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