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# Inequality and absolute value questions from my collection

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Math Expert
Joined: 02 Sep 2009
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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: inequality-and-absolute-value-questions-from-my-collection-86939-20.html#p653690

2. If y is an integer and $$y = |x| + x$$, is $$y = 0$$?
(1) $$x < 0$$
(2) $$y < 1$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-20.html#p653695

3. Is $$x^2 + y^2 > 4a$$?
(1) $$(x + y)^2 = 9a$$
(2) $$(x – y)^2 = a$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653697

4. Are x and y both positive?
(1) $$2x-2y=1$$
(2) $$\frac{x}{y}>1$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653709

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

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653731

6. If x and y are integer, is y > 0?
(1) $$x +1 > 0$$
(2) $$xy > 0$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653740

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

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653783 AND inequality-and-absolute-value-questions-from-my-collection-86939-160.html#p1111747

8. $$a*b \neq 0$$. Is $$\frac{|a|}{|b|}=\frac{a}{b}$$?
(1) $$|a*b|=a*b$$
(2) $$\frac{|a|}{|b|}=|\frac{a}{b}|$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653789

9. Is n<0?
(1) $$-n=|-n|$$
(2) $$n^2=16$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653792

10. If n is not equal to 0, is |n| < 4 ?
(1) $$n^2 > 16$$
(2) $$\frac{1}{|n|} > n$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653796

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

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653853

12. Is r=s?
(1) $$-s \leq r \leq s$$
(2) $$|r| \geq s$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653870

13. Is $$|x-1| < 1$$?
(1) $$(x-1)^2 \leq 1$$
(2) $$x^2 - 1 > 0$$

Solution: inequality-and-absolute-value-questions-from-my-collection-86939-40.html#p653886

Official answers (OA's) and detailed solutions are in my posts on pages 2 and 3.
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Re: Inequality and absolute value questions from my collection [#permalink]

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23 Oct 2014, 01:21
Bunuel wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

hi
m a little confused here. How is A sufficient

isn't n^2>16 ----> n^2-16>0 ---> (N^2-4^2) >0 ----> n+4>0 or n-4>0 ----> n> -4 or n>4.....how did u get n<-4....

kindly correct me if m wrong

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

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23 Oct 2014, 01:23
sugand wrote:
Bunuel wrote:
10. If n is not equal to 0, is |n| < 4 ?
(1) n^2 > 16
(2) 1/|n| > n

Question basically asks is -4<n<4 true.

(1) n^2>16 --> n>4 or n<-4, the answer to the question is NO. Sufficient.

(2) 1/|n| > n, this is true for all negative values of n, hence we can not answer the question. Not sufficient.

hi
m a little confused here. How is A sufficient

isn't n^2>16 ----> n^2-16>0 ---> (N^2-4^2) >0 ----> n+4>0 or n-4>0 ----> n> -4 or n>4.....how did u get n<-4....

kindly correct me if m wrong

n^2 > 16 means |n| > 4, which means that n > 4 or n < -4.

Notice that n > -4 or n > 4 does not make any sense. What are the possible value of n in this case?
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Re: Inequality and absolute value questions from my collection [#permalink]

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11 Nov 2014, 07:04
Bunuel wrote:
13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

I was hoping if we could compare the above question (in terms of asked ranges and the range that we calculate) with the below one:

Bunuel wrote:
earnit wrote:
Thoughtosphere wrote:
If x is not equal to 0, is |x| less than 1?

(1) x/|x|< x

(2) |x| > x

Will really appreciate if answer is supported by explanation.

1 - x / |x| < x This would mean that -1 < x < 0 & 0 < x <infinite
Thus 1 is insufficient

2 - |x| > x - This would mean that negative infinite values < x < 0 This is also not sufficient.

Combining 1 & 2 - the overlapping region is -1 < x < 0

We can conclude that |x| < 1

So C

Solving the Question is not much of a problem as deciding whether our solved range: $$-1 < x < 0$$ is SUFFICIENT to answer the asked range: $$-1 < x < 1$$

I need to understand that: Is our answer sufficient to conclude that YES x lies between (-1,1) even though our solution was that x lies between (-1,0) ?

The question asks whether -1 < x< 1. We got that -1 < x < 0. So, the answer is YES: x is definitely from the range (-1, 1).

WHAT IS THE UNDERLYING REASON FOR DISCARDING A GIVEN RANGE IN ONE QUESTION AND A GIVEN RANGE IN OTHER QUESTION?

Seems to be a Classic case of DS.

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

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13 Nov 2014, 06:21
Marco83 wrote:
h2polo wrote:
4. Are x and y both positive?
(1) 2x-2y=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: 2x-2y=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

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

stmt1--> 2x-2y = 1

x-y=1/2
(1)-(1/2)=1/2 {both x and y positive}
(-1/2)-(-1)=1/2 {both x and y negative}
Not Sufficient

stmt2--> x/y >1
if this is true then both x and y has to be of same sign.
Not Sufficient

combining 1 n 2: condition x/y > 1 will eliminate the (-1/2)-(-1)=1/2 {both x and y negative values} as (-1/2)/(-1) is not greater than 1
thus we left with (1)-(1/2)=1/2 {both x and y positive}
hence [C]
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Re: Inequality and absolute value questions from my collection [#permalink]

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09 Jan 2015, 14:34
Hi Bunuel,

Many Thanks

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.

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=0.
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 nonnegative only. Hence if xy is negative 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.

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

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11 Jan 2015, 11:35
bhatiavai wrote:
Hi Bunuel,

Many Thanks

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.

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=0.
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 nonnegative only. Hence if xy is negative 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.

Easier approach is given here: inequality-and-absolute-value-questions-from-my-collection-86939-160.html#p1111747

Hope it helps.
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Re: Inequality and absolute value questions from my collection [#permalink]

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09 Feb 2015, 00:21
4. Are x and y both positive?
(1) 2x-2y=1
(2) x/y>1

S1- 2(x-y)=1, x-y=1/2
X could be +ve and y -ve. Or the obvious case: x and y could both be +ve. AD
S2- x and y have same signs. Both could be -ve and still have a fraction bigger than 1. B

S1+S2- X and Y cant both be -ve, because, though it would comply with the 2nd statements restrictions, it wouldn't with the 1st. i.e, inserted in S1's equation, the RHS would be a -ve number not 1/2. So only left option is that both be +ve. E

C it is then

Cheerio

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

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09 Feb 2015, 08:14
Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

Great question.
I looked at S1 and as usual thought of special numbers, but then proceeding with -0=|-0| did not make sense to me because I thought "0 is neither +ve nor -ve so why negate it and then find absolute value". Is there some logic i'm missing or is it just a 'possible calculation'?
Thank you

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

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09 Feb 2015, 08:17
deeuk wrote:
Bunuel wrote:
9. Is n<0?
(1) -n=|-n|
(2) n^2=16

Great question.
But -0=|-0| did not make sense to me because I thought "0 is neither +ve nor -ve so why negate it and then find absolute value". I did not know that was a possible direction. Is there some logic i'm missing or is it just a 'possible calculation'?
Thank you

Yes, 0 is neither negative nor positive but there is nothing wrong in writing -0, or |0|, because -0=0 and |0| = 0.
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Re: Inequality and absolute value questions from my collection [#permalink]

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09 Feb 2015, 10:15
Bunuel wrote:
13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

About S1: can i say |x-1|<=1 after rooting the given and putting it in modulus? I recall reading about squaring being used to remove modulus, so I thought to reverse that. Please correct me.

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

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10 Feb 2015, 06:00
deeuk wrote:
Bunuel wrote:
13. Is |x-1| < 1?
(1) (x-1)^2 <= 1
(2) x^2 - 1 > 0

Last one.

Is |x-1| < 1? Basically the question asks is 0<x<2 true?

(1) (x-1)^2 <= 1 --> x^2-2x<=0 --> x(x-2)<=0 --> 0<=x<=2. x is in the range (0,2) inclusive. This is the trick here. x can be 0 or 2! Else it would be sufficient. So not sufficient.

(2) x^2 - 1 > 0 --> x<-1 or x>1. Not sufficient.

(1)+(2) Intersection of the ranges from 1 and 2 is 1<x<=2. Again 2 is included in the range, thus as x can be 2, we can not say for sure that 0<x<2 is true. Not sufficient.

About S1: can i say |x-1|<=1 after rooting the given and putting it in modulus? I recall reading about squaring being used to remove modulus, so I thought to reverse that. Please correct me.

Yes, that;s correct. We CAN take the square root from both sides of (x-1)^2 <= 1 because both sides are non-negative.

For more on inequalities check here: inequalities-tips-and-hints-175001.html

Hope it helps.
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Re: Inequality and absolute value questions from my collection [#permalink]

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29 May 2015, 09:51
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.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?

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

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30 May 2015, 04:01
sinhap07 wrote:
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.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?

First of all |-n| = |n|, so we are given that -n = |n|, which implies that n <= 0.
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Re: Inequality and absolute value questions from my collection [#permalink]

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30 May 2015, 04:15
Bunuel wrote:
sinhap07 wrote:
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.

Bunuel

Cant get thru stmt 1. Opening the mod we get either -n=-n or -n=n. First case gives null so am left with n=0 and hence sufficient. Where am I wrong in opening up the mod.?

First of all |-n| = |n|, so we are given that -n = |n|, which implies that n <= 0.

How did we arrive at |-n| = |n|?

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

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30 May 2015, 04:17
sinhap07 wrote:
How did we arrive at |-n| = |n|?

It's true for ANY x that |-x| = |x|: |-1| = |1|.
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Re: Inequality and absolute value questions from my collection [#permalink]

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01 Jun 2015, 01:35
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 $$6*x*y = x^2*y + 9*y$$:

$$y*(x^2-6x+9)=0$$ --> $$y*(x-3)^2=0$$. 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) $$y-x=3$$. If y is not 0, x must be 3 and y-x 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.

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*(x-3)^2 = 0.......(1)

Statement [1] y-x=3 or y = x+3
sub'ing y in (1)
(x+3)(x-3)(x-3) = 0
(x^2-9)(x-3)=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 (x-3)^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?
Thanks!

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

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01 Jun 2015, 01:59
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 $$6*x*y = x^2*y + 9*y$$:

$$y*(x^2-6x+9)=0$$ --> $$y*(x-3)^2=0$$. 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) $$y-x=3$$. If y is not 0, x must be 3 and y-x 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.

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*(x-3)^2 = 0.......(1)

Statement [1] y-x=3 or y = x+3
sub'ing y in (1)
(x+3)(x-3)(x-3) = 0
(x^2-9)(x-3)=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 (x-3)^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?
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 [#permalink]

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01 Jun 2015, 02:27
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^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.

Bunuel,
Thanks for keeping me honest
After adding [1] + [2] to get x^2 + y^2 = 5a, I jumped off the cliff with Answer = C

If the question has stated explicitly that x,y,a are Non zero....the answer would have been C ?

Many thanks!

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

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01 Jun 2015, 02:43
rohitd80 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^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.

Bunuel,
Thanks for keeping me honest
After adding [1] + [2] to get x^2 + y^2 = 5a, I jumped off the cliff with Answer = C

If the question has stated explicitly that x,y,a are Non zero....the answer would have been C ?

Many thanks!

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

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01 Jun 2015, 20:13
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^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.

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,

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Re: Inequality and absolute value questions from my collection   [#permalink] 01 Jun 2015, 20:13

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