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

81

This post received KUDOS

Expert's post

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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]
24 Jan 2012, 08:01

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.

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. (...)

Bunuel, how did you figure out that |x+2|=|y+2| can take only two possible forms? I did it by evaualting four scenarios when I combine these possibilities: x+2>0, x+2<0, y+2>0, y+2<'0 Is there a faster method to do it? I tried by using the method proposed by "walker", but I think that it doesn't work when there are more than one variable. Please your help. Thanks! _________________

"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."

Re: Inequality and absolute value questions from my collection [#permalink]
24 Jan 2012, 12:47

Expert's post

metallicafan 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.

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. (...)

Bunuel, how did you figure out that |x+2|=|y+2| can take only two possible forms? I did it by evaualting four scenarios when I combine these possibilities: x+2>0, x+2<0, y+2>0, y+2<'0 Is there a faster method to do it? I tried by using the method proposed by "walker", but I think that it doesn't work when there are more than one variable. Please your help. Thanks!

We have \(|x+2|=|y+2|\). If both absolute values expand with + or - sign we'll get: \(x+2=y+2\) (notice that \(-(x+2)=-(y+2)\) is exactly the same); If they will expand with different signs we'll get: \(-(x+2)=y+2\) (notice that \(x+2=-(y+2)\) is exactly the same).

Welcome to GMAT Club. Let me assist you with this set of problems: first post on each page contains links to the detailed solutions of the questions with OA's.

Re: Inequality and absolute value questions from my collection [#permalink]
23 Feb 2012, 10:41

Quote:

4. Are x and y both positive? (1) 2x-2y=1 (2) x/y>1

(1) 2x-2y=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. 2x-2y=1 --> y=x-1/2, 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.

In the first statement from 2x-2y=1 --> we can sat x-y=1/2

So it cud be 8.5-8 or 0.25 - (-0.25) HOw can we say both x and y are positive?

similarily statement 2

x/y>1 =>x>y

how can we be sure x and y have the same sign we can have 8>7 or 8>-8

Bunel can you pls xplain..or am i missing sumtin fundamental?

Re: Inequality and absolute value questions from my collection [#permalink]
25 Feb 2012, 06:51

Quote:

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.

Answer: D.

when we have x+y=-4 Acc to the explanation , we have This will occur when either x or y is less then -2 and the other is more than -2

so x can be be -3,-4,-5..(X<-2) and y can be -1,0,1...(y>-2)

a solution could that -2-2=-4

in this case it does not satisfy ? May be this is trivial but your clarification will help me correct my assumption

Re: Inequality and absolute value questions from my collection [#permalink]
25 Feb 2012, 08:06

Expert's post

shankar245 wrote:

Quote:

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.

Answer: D.

when we have x+y=-4 Acc to the explanation , we have This will occur when either x or y is less then -2 and the other is more than -2

so x can be be -3,-4,-5..(X<-2) and y can be -1,0,1...(y>-2)

a solution could that -2-2=-4

in this case it does not satisfy ? May be this is trivial but your clarification will help me correct my assumption

Thanks in advance.

When we consider ranges to expand an absolute value we should put equal sign (=) in either of the range. For our question we put equal sign for the first range (case A) when we are analyzing the case when x and y are both \(\geq{-2}\) than -2 OR both \(\leq{-2}\). So the scenario when \(x=y=-2\) (\(x+y=-2-2=-4\)) is included in case A.

Re: Inequality and absolute value questions from my collection [#permalink]
02 Apr 2012, 19:09

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.

Answer should be A.

for evaluating 2nd Option- y<1, try with value x= 0.25, then Y would be |0.25|+0.25 = 0.5, which is less than 1. It is not mentioned that x is also an integer.

And also as you mentioned for negative value of x, it will be equal to zero. So it doesnt give definite ans whether y=0. So option (2) not sufficient.

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

Expert's post

roshan1985 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.

Answer should be A.

for evaluating 2nd Option- y<1, try with value x= 0.25, then Y would be |0.25|+0.25 = 0.5, which is less than 1. It is not mentioned that x is also an integer.

And also as you mentioned for negative value of x, it will be equal to zero. So it doesnt give definite ans whether y=0. So option (2) not sufficient.

Welcome to GMAT Club.

The red part in your reasoning is not correct: y cannot be 0.25 since given that y is an integer. So, answer D is correct.

Re: Inequality and absolute value questions from my collection [#permalink]
03 Apr 2012, 06:47

h2polo 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

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 = (x-3)^2 x = 3

Statement 1:

y-x=3 y-3=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.

solved the same. a doubt! when this is a DS , cant we take statement 1 with 2 variables 'x' and 'y' will give the answer for xy? but the statement 2 wont.

Re: Inequality and absolute value questions from my collection [#permalink]
03 Apr 2012, 09:08

Expert's post

kashishh wrote:

h2polo 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

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 = (x-3)^2 x = 3

Statement 1:

y-x=3 y-3=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.

solved the same. a doubt! when this is a DS , cant we take statement 1 with 2 variables 'x' and 'y' will give the answer for xy? but the statement 2 wont.

Re: Inequality and absolute value questions from my collection [#permalink]
20 Jun 2012, 17:43

sriharimurthy wrote:

Quote:

12. Is r=s? (1) -s<=r<=s (2) |r|>=s

St. (1) : -s <= r < = s Clearly Insufficient.

St. (2) : |r| >= s When r > 0 ; r >= s. When r < 0 ; -r >= s ; r <= -s Therefore, this statement can be rewritten as : -s >= r >= s Insufficient.

St. (1) and (2) : -s <= r < = s ; -s >= r >= s For both statements to be simultaneously valid, r must be equal to s. Hence Sufficient.

Answer : C

hi sriharimurthi,

If we have the statements -s <= r < = s and -s >= r >= s, there could be 2 possible answers; r=s or r=-s. Both the possible answers satisfy the 2 equations/inequalities. So isn't the answer E and not C?

Am I missing something. Could you please help? Thanks v much.

Re: Inequality and absolute value questions from my collection [#permalink]
21 Jun 2012, 00:13

Expert's post

shivamayam wrote:

sriharimurthy wrote:

Quote:

12. Is r=s? (1) -s<=r<=s (2) |r|>=s

St. (1) : -s <= r < = s Clearly Insufficient.

St. (2) : |r| >= s When r > 0 ; r >= s. When r < 0 ; -r >= s ; r <= -s Therefore, this statement can be rewritten as : -s >= r >= s Insufficient.

St. (1) and (2) : -s <= r < = s ; -s >= r >= s For both statements to be simultaneously valid, r must be equal to s. Hence Sufficient.

Answer : C

hi sriharimurthi,

If we have the statements -s <= r < = s and -s >= r >= s, there could be 2 possible answers; r=s or r=-s. Both the possible answers satisfy the 2 equations/inequalities. So isn't the answer E and not C?

Am I missing something. Could you please help? Thanks v much.

Sriharimurthi solution is not correct. OA for this question is E, not C. The links to the OA's and solutions are given in the initial post (if the links does not work, then switch view mode of the topic from "Best Reply" to "Oldest" and the links from the initial post inequality-and-absolute-value-questions-from-my-collection-86939-160.html#p652806 will lead you to the solutions).

As for this question: 12. Is r=s?

(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.

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