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Inequality and absolute value questions from my collection [#permalink]
16 Nov 2009, 10:33

86

This post received KUDOS

Expert's post

274

This post was BOOKMARKED

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]
11 Oct 2012, 02:11

Bunuel wrote:

11. Is |x+y|>|x-y|? (1) |x| > |y| (2) |x-y| < |x|

To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2 OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.

(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Answer: B.

Hi Bunuel - Can this solved in the below way?

Is |x+y|>|x-y|?

Since both sides are +ve we can square both side of the inequality.... On squaring we get xy>0?

statement 1

(1) |x| > |y|

This is NS as xy can be opp sign as well as same sign

(2) |x-y| < |x|

Squaring on both sides we get y^2 < 2xy Y cannot be zero otherwise the inequality cannot hold so Y^2 is +ve hence xy is +ve So we can answer the question xy>0

Re: Inequality and absolute value questions from my collection [#permalink]
11 Oct 2012, 02:22

Expert's post

Jp27 wrote:

Bunuel wrote:

11. Is |x+y|>|x-y|? (1) |x| > |y| (2) |x-y| < |x|

To answer this question you should visualize it. We have comparison of two absolute values. Ask yourself when |x+y| is more then than |x-y|? If and only when x and y have the same sign absolute value of x+y will always be more than absolute value of x-y. As x+y when they have the same sign will contribute to each other and x-y will not.

5+3=8 and 5-3=2 OR -5-3=-8 and -5-(-3)=-2.

So if we could somehow conclude that x and y have the same sign or not we would be able to answer the question.

(1) |x| > |y|, this tell us nothing about the signs of x and y. Not sufficient.

(2) |x-y| < |x|, says that the distance between x and y is less than distance between x and origin. This can only happen when x and y have the same sign, when they are both positive or both negative, when they are at the same side from the origin. Sufficient. (Note that vise-versa is not right, meaning that x and y can have the same sign but |x| can be less than |x-y|, but if |x|>|x-y| the only possibility is x and y to have the same sign.)

Answer: B.

Hi Bunuel - Can this solved in the below way?

Is |x+y|>|x-y|?

Since both sides are +ve we can square both side of the inequality.... On squaring we get xy>0?

statement 1

(1) |x| > |y|

This is NS as xy can be opp sign as well as same sign

(2) |x-y| < |x|

Squaring on both sides we get y^2 < 2xy Y cannot be zero otherwise the inequality cannot hold so Y^2 is +ve hence xy is +ve So we can answer the question xy>0

Re: Inequality and absolute value questions from my collection [#permalink]
23 Dec 2012, 03:40

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,

Thanks for the explanation to the above Q.

Regarding st 1 i.e X less than zero then [m]y=|x|+x = -x+x=0,

1. we know any value in modulus is positive then ideally the above should be interpreted as [m]y=|x|+x--> [m]y=x-x=0. 2.Also if from St 1 if we x<0 then [m]y=|x|+x= -x-x=-2x

3. Where as we also know that |x|= -x for X<0 and |x|= x for X>/ 0

So can you please tell me where am I going wrong with the concept.

Thanks Mridul _________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Re: Inequality and absolute value questions from my collection [#permalink]
23 Dec 2012, 04:29

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.

Hello Bunuel,

I got A as the answer to the Q.

From St1, we have -n=|-n|---> -n=n (As Mod value is +ve)---> we have 2n=0 or -2n=0. In both case we can say that n=0 and hence Ans should be A.

From your explanation, it is very clear that either n<0 or n=0. Could you tell me what was your approach to this Question. I mean did you assume values of 1. n as less than zero, 2. ngreater than zero and 3. n equal to zero

and check under which condition the St1 holds true.

If so, would this be a standard way of doing a modulus Question because clearly I just considered only 1 of the above conditions here.

Your inputs please

Thanks Mridul _________________

“If you can't fly then run, if you can't run then walk, if you can't walk then crawl, but whatever you do you have to keep moving forward.”

Re: Inequality and absolute value questions from my collection [#permalink]
23 Dec 2012, 04:37

Expert's post

mridulparashar1 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,

Thanks for the explanation to the above Q.

Regarding st 1 i.e X less than zero then y=|x|+x = -x+x=0,

1. we know any value in modulus is positive then ideally the above should be interpreted as y=|x|+x--> y=x-x=0. 2.Also if from St 1 if we x<0 then y=|x|+x= -x-x=-2x

3. Where as we also know that |x|= -x for X<0 and |x|= x for X>/ 0

So can you please tell me where am I going wrong with the concept.

Thanks Mridul

Absolute value properties:

When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|={-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

When \(x\geq{0}\) then \(|x|=x\), or more generally when \(some \ expression\geq{0}\) then \(|some \ expression|={some \ expression}\). For example: \(|5|=5\);

So, if \(x<0\), then \(|x|=-x\) and \(y=|x|+x=-x+x=0\).

Re: Inequality and absolute value questions from my collection [#permalink]
25 Dec 2012, 04:19

Expert's post

Bunuel wrote:

debayan222 wrote:

Great collection Bunuel...Kudos.. Are these Qs. included in your signature or they exist as separate entity?

merry Xmas...Happy Holidays.

Yes, they are in Inequalities set.

Thanks a lot Bunuel.. Well I guess, whatever Qs come from you directly to the forum, are included in you Sig. ? Hope I got you right.. _________________

(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.

I got C. Can you please explain why this is incorrect?

Statement 1 gives x^2+2xy+y^2 = 9a, and I rewrote it to 9a-2xy = x^2+y^2 Statement 2 gives x^2-2xy+y^2=a, and i rewrote this to x^2+y^2= a +2xy

Together, I have 9a-2xy = a+2xy, which leads to 8a = 4xy, but 8a is also equal to x^2+y^2? ... this last part must be wrong? _________________

Re: Inequality and absolute value questions from my collection [#permalink]
21 Feb 2013, 20:23

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, I think I need some conceptual help. Why should we not solve statement 1 by rewriting the two statements and then adding them together? (Besides the fact that it's time consuming....) I rewrote them and found 3x^2 -10 = y for the positive absolute vlaue, and -3x^2+14=y for the negative abs value. From this, I added them together and got y=4..

Can you please explain what I'm getting wrong conceptually? Thanks so much!!!! I appreciate your kindness. _________________

(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.

I got C. Can you please explain why this is incorrect?

Statement 1 gives x^2+2xy+y^2 = 9a, and I rewrote it to 9a-2xy = x^2+y^2 Statement 2 gives x^2-2xy+y^2=a, and i rewrote this to x^2+y^2= a +2xy

Together, I have 9a-2xy = a+2xy, which leads to 8a = 4xy, but 8a is also equal to x^2+y^2? ... this last part must be wrong?

The red part is not right. If you sum the two equations you'll get 2(x^2+y^2)=10a --> x^2+y^2=5a.

(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.

I got C. Can you please explain why this is incorrect?

Statement 1 gives x^2+2xy+y^2 = 9a, and I rewrote it to 9a-2xy = x^2+y^2 Statement 2 gives x^2-2xy+y^2=a, and i rewrote this to x^2+y^2= a +2xy

Together, I have 9a-2xy = a+2xy, which leads to 8a = 4xy, but 8a is also equal to x^2+y^2? ... this last part must be wrong?

The red part is not right. If you sum the two equations you'll get 2(x^2+y^2)=10a --> x^2+y^2=5a.

So I understand that you summed the equations and got that answer. But why is setting them equal to each other wrong in this case? I'm trying to figure out what concept I'm missing so that I don't end up doing it again. _________________

Re: Inequality and absolute value questions from my collection [#permalink]
22 Feb 2013, 06:26

Bunuel wrote:

JJ2014 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, I think I need some conceptual help. Why should we not solve statement 1 by rewriting the two statements and then adding them together? (Besides the fact that it's time consuming....) I rewrote them and found 3x^2 -10 = y for the positive absolute vlaue, and -3x^2+14=y for the negative abs value. From this, I added them together and got y=4..

Can you please explain what I'm getting wrong conceptually? Thanks so much!!!! I appreciate your kindness.

|x^2-4|=x^2-4 when x^2-4>0; |x^2-4|=-(x^2-4) when x^2-4<=0.

So, the two equations you'll get from the original are relevant for different ranges of x. Hence, you cannot consider them as two separate equations and solve.

To put it simply: we cannot get the single value of y from 3|x^2 -4| = y - 2. Consider y=2 and x=2 OR y=11 and x=1.

So I understand that you summed the equations and got that answer. But why is setting them equal to each other wrong in this case? I'm trying to figure out what concept I'm missing so that I don't end up doing it again.

What are you trying to get when setting "them" equal? Anyway, you won't be able to solve two equations with three unknowns. _________________

Re: Inequality and absolute value questions from my collection [#permalink]
28 Feb 2013, 05:35

7. |x+2|=|y+2| what is the value of x+y? (1) xy<0 (2) x>2 y<2

The solution seem confusing to me as I see four cases: a] x<-2, y<-2 b]x>-2, y>-2 c] x<-2, y>-2 d]x>-2, y<-2

case [a] and [b] support x=y while case [c] and [d] support x+y=-4

when xy<0, the case [c]or[d] always do not apply, for example: x=-3 and y=3 would come under case[c] and x=-1 and y=3 would come under case [b] , so it is insufficient.

when x>2 , y<2, we have a case [b] with x=3, y=-1 and a case [d] with x=3,y=-3. So insufficient

when we combine(1)+(2) , we have a case as shown above , it is also insufficient.

(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 Bunnel Solved the 1st statement like this - \((x + y)^2 = 9a\) Since \(x^2 + y^2 >= 2xy\) \(x^2 + y^2 + x^2 + y^2 >= 9a\) \(2(x^2 + y^2) >= 9a\) \(x^2 + y^2 >= 4.5a\) Now this would have been sufficient if a is not = 0 had been given in the stem Is this approach to the problem alright?? Is this st sufficient if it is given that a is not equal to 0 Thanks

(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 Bunnel Solved the 1st statement like this - \((x + y)^2 = 9a\) Since \(x^2 + y^2 >= 2xy\) \(x^2 + y^2 + x^2 + y^2 >= 9a\) \(2(x^2 + y^2) >= 9a\) \(x^2 + y^2 >= 4.5a\) Now this would have been sufficient if a is not = 0 had been given in the stem Is this approach to the problem alright?? Is this st sufficient if it is given that a is not equal to 0 Thanks

Yes, but we don't know whether x is 0. _________________

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