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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) \(x^3< 0\)

Re: Inequality and absolute value questions from my collection [#permalink]

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20 Jan 2010, 03:45

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.

Quote:

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=-2. B. x+2=-y-2 --> 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 positive only. Hence if xy is not positive we have scenario B and x+y=-4. Also note that vise-versa 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.

Wonderful question, I solved it up to an extent but in reasoning I messed up. Question for the quoted part which talks about the scenarios to reducing to two scenarios. I can think and verfy also though the validity of it, but want to drill further and understand how could we generalise this scenario if we could ?

Like what if |x+3|=|X-2| or |x+3|=|X-2| + |X+4| etc , just crude examples I have put , How could we generalise such scenarios ? instead of imagining six scenarios or more .

Re: Inequality and absolute value questions from my collection [#permalink]

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20 Jan 2010, 06:15

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.

Quote:

(1) -s<=r<=s, we can conclude two things from this statement: A. s is either positive or zero, as -s<=s;

How do you conclude A here can't make out or I am tired.; anyways, it is mental marathon .. wonderful questions Bunuel. Any more link for inequality as I need some more practise .

Re: Inequality and absolute value questions from my collection [#permalink]

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21 Jan 2010, 00:41

Expert's post

GMATMadeeasy wrote:

How do you conclude A here can't make out or I am tired.; anyways, it is mental marathon .. wonderful questions Bunuel. Any more link for inequality as I need some more practise .

We have \(-s<=r<=s\) --> \(-s<=s\). Now if \(s\) in negative, let's say \(-2\), then we would have \(-(-2)<=-2\) --> \(2<=-2\), which is not right. Hence \(s\) can not be negative. But \(s\) can be zero --> \(0<=0\), true. _________________

Re: Inequality and absolute value questions from my collection [#permalink]

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21 Jan 2010, 00:55

Expert's post

GMATMadeeasy 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.

Quote:

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=-2. B. x+2=-y-2 --> 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 positive only. Hence if xy is not positive we have scenario B and x+y=-4. Also note that vise-versa 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.

Wonderful question, I solved it up to an extent but in reasoning I messed up. Question for the quoted part which talks about the scenarios to reducing to two scenarios. I can think and verfy also though the validity of it, but want to drill further and understand how could we generalise this scenario if we could ?

Like what if |x+3|=|X-2| or |x+3|=|X-2| + |X+4| etc , just crude examples I have put , How could we generalise such scenarios ? instead of imagining six scenarios or more .

We can count the # of scenarios for the above examples as well, but as in these examples there is only one variable we can directly solve them using the ranges, so no need to use the approach we used in solving the original question.

|x+3|=|x-2|, two check points -3 and 2, thus we'll have three ranges in which we should expand the absolute values:

A. x<-3 --> -(x+3)=-(x-2) --> -3=2, which is not true, hence in this range we have no solution; B. -3<=x<=2 --> x+3=-(x-2) --> x=-1/2, which IS in the range we are expanding so it's a valid solution; C. x>2 --> x+3=x+2 --> 3=2, which is not true, hence in this range we have no solution.

The equation |x+3|=|x-2| has only one solution x=-1/2. If you look at the scenarios A,B and C you'll see that |x+3|=|x-2| can take also only TWO forms x+3=x+2 or x+3=-(x-2), but the ranges are important here and we should consider these forms in their respective ranges.

(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.

Answer: E.

Hi Bunuel,

I kind of disagree with your conclusion when you combined both the stmts. If x,y, and a all are 0 then the actual question (x^2+y^2 > 4a) itself will become whether 0 > 0 ?....so I would say that the answer should be C.

hi xyztroy,

i think i can answer ur question.

in question, no limits for x and y are given, like x&y are integers or x&y are real numbers. so x and y can assume any values, including 0. but we have to conclusively show that (x^2+y^2 > 4a). as you see, 1&2 are individually insufficient. combining 1&2 we have (x^2+y^2 = 5a), which is definitely greater than 4a. when you substitute values for x,y and a, all values of x,y and a which satisfy (x^2+y^2 = 5a) also satisfies (x^2+y^2 > 4a), except the values x=y=a=0. so two cases arise. hence insufficient.

answer:E

I think, when statements are given we have to answer whether based on statements given question can be answered, we should not question the validity of statements or a statements derived form the statements. after deriving from statement 1 and 2 we deduce that x^2+y^2 =5a, this is enough to show that the question asked x^2+y^2>4a is satisfied. we are not supposed to validate whether statement x^2+y^2 =5a holds. if u try to put x,y as 0 then just ask your self what question is trying to ask? is 0>0. so answer has to be C

Re: Inequality and absolute value questions from my collection [#permalink]

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15 Feb 2010, 06:15

Expert's post

sandeep25398 wrote:

logan wrote:

xyztroy wrote:

Hi Bunuel,

I kind of disagree with your conclusion when you combined both the stmts. If x,y, and a all are 0 then the actual question (x^2+y^2 > 4a) itself will become whether 0 > 0 ?....so I would say that the answer should be C.

hi xyztroy,

i think i can answer ur question.

in question, no limits for x and y are given, like x&y are integers or x&y are real numbers. so x and y can assume any values, including 0. but we have to conclusively show that (x^2+y^2 > 4a). as you see, 1&2 are individually insufficient. combining 1&2 we have (x^2+y^2 = 5a), which is definitely greater than 4a. when you substitute values for x,y and a, all values of x,y and a which satisfy (x^2+y^2 = 5a) also satisfies (x^2+y^2 > 4a), except the values x=y=a=0. so two cases arise. hence insufficient.

answer:E

I think, when statements are given we have to answer whether based on statements given question can be answered, we should not question the validity of statements or a statements derived form the statements. after deriving from statement 1 and 2 we deduce that x^2+y^2 =5a, this is enough to show that the question asked x^2+y^2>4a is satisfied. we are not supposed to validate whether statement x^2+y^2 =5a holds. if u try to put x,y as 0 then just ask your self what question is trying to ask? is 0>0. so answer has to be C

1. \(x=2\), \(y=1\) and \(a=1\). These values satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is YES, as \(5>4\) is true. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

2. \(x=0\), \(y=0\) and \(a=0\). These values ALSO satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is NO, as LHS is \(0\), RHS is also \(0\) and \(0>0\) is NOT TRUE. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

Two different answers. Not sufficient.

Answer: E.

So question is not asking whether \(0>0\). Question is asking whether "\(x^2+y^2>4a\) true". We got that for smoe values YES and for some values NO.

Re: Inequality and absolute value questions from my collection [#permalink]

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16 Feb 2010, 06:37

answer:E[/quote] I think, when statements are given we have to answer whether based on statements given question can be answered, we should not question the validity of statements or a statements derived form the statements. after deriving from statement 1 and 2 we deduce that x^2+y^2 =5a, this is enough to show that the question asked x^2+y^2>4a is satisfied. we are not supposed to validate whether statement x^2+y^2 =5a holds. if u try to put x,y as 0 then just ask your self what question is trying to ask? is 0>0. so answer has to be C[/quote]

1. \(x=2\), \(y=1\) and \(a=1\). These values satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is YES, as \(5>4\) is true. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

2. \(x=0\), \(y=0\) and \(a=0\). These values ALSO satisfy both statements and the answer to the question "is \(x^2+y^2>4a\) true" is NO, as LHS is \(0\), RHS is also \(0\) and \(0>0\) is NOT TRUE. (Note here that these values naturally satisfy \(x^2+y^2=5a\) too)

Two different answers. Not sufficient.

Answer: E.

So question is not asking whether \(0>0\). Question is asking whether "\(x^2+y^2>4a\) true". We got that for smoe values YES and for some values NO.

Hope it's clear.[/quote] yes Bunuel, it helped.. thank you for such amazing collections..and explanations

Re: Inequality and absolute value questions from my collection [#permalink]

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17 Feb 2010, 23:19

Bunuel wrote:

GMATMadeeasy wrote:

How do you conclude A here can't make out or I am tired.; anyways, it is mental marathon .. wonderful questions Bunuel. Any more link for inequality as I need some more practise .

We have \(-s<=r<=s\) --> \(-s<=s\). Now if \(s\) in negative, let's say \(-2\), then we would have \(-(-2)<=-2\) --> \(2<=-2\), which is not right. Hence \(s\) can not be negative. But \(s\) can be zero --> \(0<=0\), true.

You can't write it this way. -s<=r<=s means that either r lies between s & -s or r is equal to -s or r can be equal to s. But this certainly doesn't mean that you can equate -s<=s.

If I say that -4<=x<=4. This doesn't mean that I can write -4<=4.

But in any case the answer you've arrived at is correct. We can't derive anything out of the first statement as it says that -s<=r<=s. r can be anything -s or s or between -s & s.

Second says that lrl>=s ---> r>=s or r<=-s. This again is not giving any specific value.

If we combine the two we get that either r=s or r=-s. This is closer but still ambiguous. So we don't know whether r=s. Therefore, answer is E.

Re: Inequality and absolute value questions from my collection [#permalink]

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18 Feb 2010, 08:22

Expert's post

honeyrai wrote:

Bunuel wrote:

GMATMadeeasy wrote:

How do you conclude A here can't make out or I am tired.; anyways, it is mental marathon .. wonderful questions Bunuel. Any more link for inequality as I need some more practise .

We have \(-s<=r<=s\) --> \(-s<=s\). Now if \(s\) in negative, let's say \(-2\), then we would have \(-(-2)<=-2\) --> \(2<=-2\), which is not right. Hence \(s\) can not be negative. But \(s\) can be zero --> \(0<=0\), true.

You can't write it this way. -s<=r<=s means that either r lies between s & -s or r is equal to -s or r can be equal to s. But this certainly doesn't mean that you can equate -s<=s.

If I say that -4<=x<=4. This doesn't mean that I can write -4<=4.

But in any case the answer you've arrived at is correct. We can't derive anything out of the first statement as it says that -s<=r<=s. r can be anything -s or s or between -s & s.

First of all: you are excluding the possibility when -s=r=s=0.

-s<=r<=s, so -s can be equal to s, when s=0. In all other cases -s will be less than s, so there is nothing wrong in writing this as -s<=s.

Second of all: we CAN conclude one more thing from this statement: s is either positive or zero.

Here is what I wrote in my solution from page 3:

(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.

Question Stem : Is |x-1| < 1 ? When x > 1 ; x - 1 < 1 ; x < 2. When x < 1 ; -x + 1 < 1 ; x > 0. Thus it can be written as : 0 < x < 2.

St. (1) : (x-1)^2 <= 1 x^2 + 1 - 2x <= 1 x^2 - 2x <= 0 x(x - 2) <= 0 ; Thus boundary values are 0 and 2. Therefore statement can be written as : 0 <= x <= 2. Since the values are inclusive of 0 and 2, it cannot give us the answer. Insufficient.

St. (2) : x^2 - 1 > 0 (x + 1)*(x - 1) > 0 Statement can be written as x > 1 and x < -1. Thus it is possible for x to hold values which make the question stem true as well as false. Insufficient.

St. (1) and (2) : 0 <= x <= 2 ; x > 1 and x < -1 Thus combined, the statements become : 1 < x <= 2. Since it is inclusive of 2, it will give us conflicting solutions for the question stem. Hence Insufficient.

Answer : E

How do u get these boundary values. Looking at the equation, I solved it as x(x-2)<=0.... which gives x<=0 or x<=2... I know this isnt correct but can u let me know how u got... x>=0 & x <=2... Thanks _________________

Cheers! JT........... If u like my post..... payback in Kudos!!

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Question Stem : Is |x-1| < 1 ? When x > 1 ; x - 1 < 1 ; x < 2. When x < 1 ; -x + 1 < 1 ; x > 0. Thus it can be written as : 0 < x < 2.

St. (1) : (x-1)^2 <= 1 x^2 + 1 - 2x <= 1 x^2 - 2x <= 0 x(x - 2) <= 0 ; Thus boundary values are 0 and 2. Therefore statement can be written as : 0 <= x <= 2. Since the values are inclusive of 0 and 2, it cannot give us the answer. Insufficient.

St. (2) : x^2 - 1 > 0 (x + 1)*(x - 1) > 0 Statement can be written as x > 1 and x < -1. Thus it is possible for x to hold values which make the question stem true as well as false. Insufficient.

St. (1) and (2) : 0 <= x <= 2 ; x > 1 and x < -1 Thus combined, the statements become : 1 < x <= 2. Since it is inclusive of 2, it will give us conflicting solutions for the question stem. Hence Insufficient.

Answer : E

How do u get these boundary values. Looking at the equation, I solved it as x(x-2)<=0.... which gives x<=0 or x<=2... I know this isnt correct but can u let me know how u got... x>=0 & x <=2... Thanks

"Boundary values" are the roots of the equation \(x(x-2)=0\) --> \(x=0\) and \(x=2\).

I use "parabola approach" for this kind of quadratic inequalities. Basically as \(x(x-2)=0\) is an equation of upward parabola, \(x(x-2)\leq{0}\) (\(x^2-2x\leq{0}\)) is the part of the parabola, which does not lie above the X-axis. Parabola \(x(x-2)\leq{0}\) does not lie above the X-axis for the values of x in the range \(0\leq{x}\leq{2}\). If you plug the values from this range you'll see that inequality, \(x(x-2)\leq{0}\), holds true for them.

Consider inequality \((x-3)(x-7)>{0}\) (or \(x^2-10x+21>0\)). This is also upward parabola (as the coefficient of x^2 is positive), with boundary values \(x=3\) and \(x=7\). We want to determine for which value of x, this parabola is above (as we have > sign) the X-axis. The answer would be \(x<3\) and \(x>7\).

OR:

\((7+x)(3-x)>0\) (or \(-x^2-4x+21>0\)), this would be downward parabola, with boundary values \(x=-7\) and \(x=3\). This parabola lies above the X-axis for \(-7<x<3\). We can also write this as \(x^2+4x-21<0\) (or \((7+x)(x-3)<0\)), this would be upward parabola, with the same boundary values: \(x=-7\) and \(x=3\). This parabola lies below the X-axis for \(-7<x<3\), the same range as it should be.

Re: Inequality and absolute value questions from my collection [#permalink]

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04 May 2010, 12:44

Bunuel wrote:

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

(1) As we are asked to find the value of y, from this statement we can conclude only that y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative. Not sufficient.

(2) |3 - y| = 11:

y<3 --> 3-y=11 --> y=-8 y>=3 --> -3+y=11 --> y=14

Two values for y. Not sufficient.

(1)+(2) y>=2, hence y=14. Sufficient.

Answer: C.

Bunuel,

Thanks so much for all these questions and explanations!

I'm having troubles to understand the quoted question. I don't understand your explanation for statement 1, it seems like a great way to solve this problem quickly but I really just don't understand how you get there.

From this 3|x^2 -4| = y - 2 how can you conclude that y must be >=2 ?

Re: Inequality and absolute value questions from my collection [#permalink]

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04 May 2010, 13:30

Expert's post

nifoui wrote:

Bunuel wrote:

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

(1) As we are asked to find the value of y, from this statement we can conclude only that y>=2, as LHS is absolute value which is never negative, hence RHS als can not be negative. Not sufficient.

(2) |3 - y| = 11:

y<3 --> 3-y=11 --> y=-8 y>=3 --> -3+y=11 --> y=14

Two values for y. Not sufficient.

(1)+(2) y>=2, hence y=14. Sufficient.

Answer: C.

Bunuel,

Thanks so much for all these questions and explanations!

I'm having troubles to understand the quoted question. I don't understand your explanation for statement 1, it seems like a great way to solve this problem quickly but I really just don't understand how you get there.

From this 3|x^2 -4| = y - 2 how can you conclude that y must be >=2 ?

Thx for the help!

\(3|x^2 -4| = y - 2\) --> Left hand side (LHS) is an expression with absolute value, absolute value is never negative (\(|expression|\geq{0}\)) --> \(3|x^2-4|\geq{0}\). So as \(LHS\geq{0}\), then RHS also must be more than zero (\(RHS\geq{0}\)) --> \(y-2\geq{0}\) --> \(y\geq{2}\).

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