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Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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27 May 2012, 08:35

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|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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28 May 2012, 09:09

4

This post received KUDOS

Expert's post

1

This post was BOOKMARKED

kashishh wrote:

Bunuel wrote:

|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Answer: D.

Hope it's clear.

Dear Bunuel,

whenever absolute value is analysed, we take two scenarios of <0 and >0. So, why the same is not considered for |x| ?

If \(x\leq{0}\) and \(y\leq{0}\) then \(|x|=|2y|\) expands as \(-x=-2y\) --> \(x=2y\); If \(x\leq{0}\) and \(y>{0}\) then \(|x|=|2y|\) expands as \(-x=2y\) --> \(x=-2y\); If \(x>{0}\) and \(y\leq{0}\) then \(|x|=|2y|\) expands as \(x=-2y\); If \(x>{0}\) and \(y>{0}\) then \(|x|=|2y|\) expands as \(x=2y\).

So as you can see \(|x|=|2y|\) can expand only in two ways \(x=2y\) or \(x=-2y\) (as shown above ++ and -- are the same, and +- and -+ are the same). _________________

Am stuck with solving the statement 1 with case scenarios. Somebody please explain your entire solutions especially the statement one positive negative scenarios.

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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28 May 2012, 08:57

Bunuel wrote:

|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Answer: D.

Hope it's clear.

Dear Bunuel,

whenever absolute value is analysed, we take two scenarios of <0 and >0. So, why the same is not considered for |x| ?

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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07 Jun 2012, 14:35

Expert's post

BDSunDevil wrote:

Bunuel: can we rewrite |x|=|2y| as x^2-4x^2=0 ? I have solved the problem doing so, but not sure if it algebraically correct. Below what i did:

(x-2y)(x+2y)=0

Using statement 1: (x-2y)*6=0 so, (x-2y)=0. Sufficient

Using statement 2: x=2y [same sign] (x-2y)=0. Sufficient

D

Yes, you can square \(|x|=|2y|\) and write \(x^2=4y^2\) --> \((x-2y)(x+2y)=0\) --> either \(x=2y\) or \(x=-2y\) the same two options as in my solution above. _________________

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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24 Jul 2012, 16:10

Bunuel wrote:

BDSunDevil wrote:

Bunuel: can we rewrite |x|=|2y| as x^2-4x^2=0 ? I have solved the problem doing so, but not sure if it algebraically correct. Below what i did:

(x-2y)(x+2y)=0

Using statement 1: (x-2y)*6=0 so, (x-2y)=0. Sufficient

Using statement 2: x=2y [same sign] (x-2y)=0. Sufficient

D

Yes, you can square \(|x|=|2y|\) and write \(x^2=4y^2\) --> \((x-2y)(x+2y)=0\) --> either \(x=2y\) or \(x=-2y\) the same two options as in my solution above.

Hi Bunuel,

I had a query regarding an official statement in the solution to this problem. Actually, the book says that , as, x+2y=6 , so a positive sum indicates that both x and 2y must be positive. However, -4+10= 10+(-4) = 6 =positive sum [both x and 2y are not positive] 10+4=14= positive sum [both x & 2y are positive] isn't it? Please clarify the confusion here..

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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27 May 2013, 15:04

Hello, I am a bit confused regarding absolute value.

If \(|x|=|2y|\), then why why aren't x and 2y both positive? If the abs. value of something (i.e. it's positive value) is equal to something else, doesn't that imply that they are both positive? For example, if x=|2y| doesn't that mean that x is positive?

Also, for #2, xy both have the same signs. If x and y are negative, why would |x|=|2y| become -x = -2y? I get that it's equal to |x|=|2y| but why even take that step? |-x| = |-2y| will always be positive, right?

Bunuel wrote:

|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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27 May 2013, 15:22

Expert's post

WholeLottaLove wrote:

Hello, I am a bit confused regarding absolute value.

If \(|x|=|2y|\), then why why aren't x and 2y both positive? If the abs. value of something (i.e. it's positive value) is equal to something else, doesn't that imply that they are both positive? For example, if x=|2y| doesn't that mean that x is positive?

Also, for #2, xy both have the same signs. If x and y are negative, why would |x|=|2y| become -x = -2y? I get that it's equal to |x|=|2y| but why even take that step? |-x| = |-2y| will always be positive, right?

Bunuel wrote:

|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Answer: D.

Hope it's clear.

The absolute value cannot be negative \(|some \ expression|\geq{0}\), or \(|x|\geq{0}\) (absolute value of x, |x|, is the distance between point x on a number line and zero, and the distance cannot be negative).

So, if given that \(x=|2y|\) then \(x\) must be more than or equal to zero (RHS is non-negative thus LHS must also be non-negative).

But in our case we have that \(|x|=|2y|\). In this case \(x\) and/or \(y\) could be negative. For, example \(x=-2\) and \(y=-1\) --> \(|x|=2=|2y|\).

As for (2): 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 \(x<0\) and \(y<0\), then \(|x|=-x\) and \(|2y|=-2y\) --> \(-x=-2y\) --> \(x=2y\). If \(x>0\) and \(y>0\), then \(|x|=x\) and \(|2y|=2y\) --> \(x=2y\), the same as in the first case.

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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27 May 2013, 16:16

Ok, so I get that Abs. value cannot be negative...distance cannot have a negative value.

We are trying to solve for x-2y, so naturally we are trying to determine x-2y. So,

If x=2y then the value of x-2y = 2y-2y = 0 OR If x=-2y (the absolute value of 2y) then the value of x-2y = -2y-2y = -4y, correct?

I guess what throws me off is when you write

When x\leq{0} then |x|=-x. What you're saying is that, for example, |-4| = -(-4) or |-4| = 4. What is the point of writing |-4| = -(-4)

One final thing...In the stem you derived x=2y, x=-2y. Okay, but in #2. one of the cases is xy>0 so we could have -x and -y. If x and y are negative, doesn't that mean that you would substitute -x and y in to get -x=-2(-y) = -x=2y?

I'm sorry for being such a dolt. Sometimes, concepts that I know are very simple are extremely difficult to understand.

Bunuel wrote:

WholeLottaLove wrote:

Hello, I am a bit confused regarding absolute value.

If \(|x|=|2y|\), then why why aren't x and 2y both positive? If the abs. value of something (i.e. it's positive value) is equal to something else, doesn't that imply that they are both positive? For example, if x=|2y| doesn't that mean that x is positive?

Also, for #2, xy both have the same signs. If x and y are negative, why would |x|=|2y| become -x = -2y? I get that it's equal to |x|=|2y| but why even take that step? |-x| = |-2y| will always be positive, right?

Bunuel wrote:

|x|=|2y|, what is the value of x-2y?

First of all \(|x|=|2y|\) means that either \(x=2y\) or \(x=-2y\).

(1) x+2y = 6. Now, the second case is not possible since if \(x=-2y\) then from this statement we would have that \(-2y+2y=6\) --> \(0=6\), which obviously is not true. So, we have that \(x=2y\), in this case \(x-2y=2y-2y=0\). Sufficient.

(2) xy>0 --> \(x\) and \(y\) are either both positive or both negative, in any case \(|x|=|2y|\) becomes \(x=2y\) (if \(x\) and \(y\) are both negative then \(|x|=|2y|\) becomes \(-x=-2y\) which is the same as \(x=2y\)). Now, if \(x=2y\) then \(x-2y=2y-2y=0\). Sufficient.

Answer: D.

Hope it's clear.

The absolute value cannot be negative \(|some \ expression|\geq{0}\), or \(|x|\geq{0}\) (absolute value of x, |x|, is the distance between point x on a number line and zero, and the distance cannot be negative).

So, if given that \(x=|2y|\) then \(x\) must be more than or equal to zero (RHS is non-negative thus LHS must also be non-negative).

But in our case we have that \(|x|=|2y|\). In this case \(x\) and/or \(y\) could be negative. For, example \(x=-2\) and \(y=-1\) --> \(|x|=2=|2y|\).

As for (2): When \(x\leq{0}\) then \(|x|=-x\), or more generally when \(some \ expression\leq{0}\) then \(|some \ expression|\leq{-(some \ expression)}\). For example: \(|-5|=5=-(-5)\);

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

So, if \(x<0\) and \(y<0\), then \(|x|=-x\) and \(|2y|=-2y\) --> \(-x=-2y\) --> \(x=2y\). If \(x>0\) and \(y>0\), then \(|x|=x\) and \(|2y|=2y\) --> \(x=2y\), the same as in the first case.

Re: |x|=|2y|, what is the value of x-2y? [#permalink]

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