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

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|x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2012, 07:18
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|x|=|2y|, what is the value of x-2y?

(1) x+2y = 6
(2) xy>0

[Reveal] Spoiler:
i wish to have clarification on st. 1.
x+2y = 6
if x = 2, y = 2 or
if x= -2 , y = 4 then also it is '6'

do we need to keep the constraint +x = +2y while evaluating st.1 ?
[Reveal] Spoiler: OA
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2012, 07:35
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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.

Hope it's clear.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  28 May 2012, 08:09
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Expert's post
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.

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).
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  05 Jun 2012, 12:18
2
KUDOS
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
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Re: IxI = I2yI what is the value of x - 2y? [#permalink]  26 Jan 2013, 11:08
1
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alexpavlos wrote:
IxI = I2yI what is the value of x - 2y?

1) x + 2y = 6
2) xy > 0

I can understand what to do with statement 2. Statement 1, I have no clue what to do with it!

Thanks!
Alex

x + 2y = 6
Hence we know that x is not equal to -2y, but |x| = |2y|
So, x = 2y
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|x|=|2y|, what is the value of x-2y? [#permalink]  08 Jun 2015, 23:21
1
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mpingo wrote:
If |x|=|2y|
what is the value of x-2y?

1. x+2y=6
2. xy>0

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

Thanks.

Hello mpingo
This topic discussed here:
x-2y-what-is-the-value-of-x-2y-133307.html

If after reading discussions above, you will have questions, than write them here and I will be glad to help.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  28 May 2012, 07:57
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.

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| ?
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  31 May 2012, 19:45
Tricky question.... I gave 2 much time to evaluate stmt 1 and went with A.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  07 Jun 2012, 13:35
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.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  24 Jul 2012, 15:10
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?
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2013, 14:04
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.

Hope it's clear.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2013, 14:22
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.

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.

For more check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html

DS questions on absolute value to practice: search.php?search_id=tag&tag_id=37
PS questions on absolute value to practice: search.php?search_id=tag&tag_id=58

Tough absolute value and inequity questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2013, 15:16
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.

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.

For more check Absolute Value chapter of Math Book: math-absolute-value-modulus-86462.html

DS questions on absolute value to practice: search.php?search_id=tag&tag_id=37
PS questions on absolute value to practice: search.php?search_id=tag&tag_id=58

Tough absolute value and inequity questions with detailed solutions: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope it helps.

Last edited by WholeLottaLove on 27 May 2013, 15:28, edited 1 time in total.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  27 May 2013, 15:26
Expert's post
WholeLottaLove wrote:
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)

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

Yes, that's correct: if x=2y, then x-2y=0 and if x=-2y, then x-2y=-4y.

As for the red part: it's just an example of the statement that if $$x\leq{0}$$ then $$|x|=-x$$.
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  29 May 2013, 02:44
|x|=|2y|, what is the value of x-2y?

(1) x+2y = 6
(2) xy>0

1) that means that x=3 and 2y=3, so difference is only 0
2) that means that x and y is not 0 and both positive or negative and x=2y, so 2y-2y=0

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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  30 Jun 2013, 11:28
|x|=|2y|, what is the value of x-2y?

x=2y
OR
x=-2y

(1) x+2y = 6

2y+2y = 6
4y = 6
y=3/2

x+2(3/2) = 6
x+3 = 6
x=3

OR
-2y+2y = 6
0=6 (Invalid...6 cannot equal 0)
With only one valid solution for x and y we can solve for x-2y.
SUFFICIENT

(2) xy>0

xy>0 means that BOTH x and y are positive or BOTH x and y are negative.
We can choose numbers to make this easier:
x=2, y=1

If x=2y, then 2=2(1)
OR
Id x=-2y, then -2 = 2(-1)

If x and y are both positive: x-2y ===> 2-2(1) = 0
If x and y are both negative: x-2y ===> -2 - 2(-1) ===> -2+2 = 0

SUFFICIENT
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Re: |x|=|2y|, what is the value of x-2y? [#permalink]  25 Jul 2014, 07:29
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If |x|=|2y|
what is the value of x-2y?

1. x+2y=6
2. xy>0

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

Thanks.
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