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

kashishh

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27 May 2012, 08:18

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

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

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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 ?

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Bunuel

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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.

Answer: D.

Hope it's clear.

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28 May 2012, 09:09

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kashishh

Bunuel

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).

General Discussion

kashishh

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28 May 2012, 08:57

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Bunuel

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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31 May 2012, 20:45

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Tricky question.... I gave 2 much time to evaluate stmt 1 and went with A.

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05 Jun 2012, 13:18

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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

Bunuel

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07 Jun 2012, 14:35

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BDSunDevil

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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24 Jul 2012, 16:10

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Bunuel

BDSunDevil

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..

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26 Jan 2013, 12:08

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alexpavlos

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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27 May 2013, 15:04

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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

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27 May 2013, 15:22

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WholeLottaLove

Bunuel

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.

WholeLottaLove

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Updated on: 27 May 2013, 16:28

Originally posted by WholeLottaLove on 27 May 2013, 16:16.
Last edited by WholeLottaLove on 27 May 2013, 16:28, edited 1 time in total.

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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

WholeLottaLove

Bunuel

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.

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27 May 2013, 16:26

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WholeLottaLove

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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29 May 2013, 03:44

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|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

D

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30 Jun 2013, 12:28

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|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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08 Jun 2015, 20:11

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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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mpingo

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

Please, use search before posting

Also, please, read rule of posting #3 about naming of topic
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If after reading discussions above, you will have questions, than write them here and I will be glad to help.

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ok, x= 2y, or x= -2y---> x-2y =0 or x+2y =0
1. x+2y =6 so x-2y =0
2. x,y >0--> x+2y can't be 0 , so x-2y =0
D

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17 May 2021, 09:23

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kashishh

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

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

Show 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 ?

|something| = |something| ----> for constructs like these square both sides and remove the Mod sign.

Stem:
$|x|=|2y|$ ----> $(x)^2= (2y)^2$--->$ x^2-(2y)^2=0$--- > is of the form $a^2-b^2$= $(a-b) * (a+b)$

stm1>$x+2y = 6$

from the equation we formed in the stem: $x^2-(2y)^2=0 $----> $(x-2y)*(x+2y)=0$

put $x+2y=6$ in the above equation, and we get $x-2y=0$

Suff.

(2) xy>0 (means both x and y are of the same sign, both +ve or -ve)

$x^2-(2y)^2=0-$---> $(x+2y)* (x-2y)=0$ ----> $x-2y= 0$. Sufficient.

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