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# If |x| - |y| = |x+y| and xy does not equal to 0, which of

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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  30 May 2013, 15:40
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I get the mechanics of flipping the signs when y is negative, but I guess I don't understand the logic.

If I take an absolute value of a number (always positive) then subtract from it an absolute value of a smaller number, which is negative how does that end up being X+Y? Are we looking just for the values of x and y, as opposed to the values of |x|-|y|?
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  30 May 2013, 17:07
If |x| - |y| = |x+y| and xy does not equal to 0, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y > 0
D. xy > 0
E. xy < 0

xy not equal to 0 means neither of them is zero , the given equation can't hold if both are positive because in that case you can remove the mod sign and it will give you y=0 which is not possible

if both x,y are positive ,x - y = x + y

Same goes for x and y both being negative .

However, one of them can be neg and one positive , x=-2 and y = 1 satisfies the equation and hence ans is E
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  30 May 2013, 22:47
Expert's post
WholeLottaLove wrote:
Yes, but why does -2|xy| = 2xy?

and is this example posted by Bunuel...
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.

How does |x| - |y| = x+y? wouldn't it be x-y?

|x|-|y| will always be (x-y) ONLY for non-negative values of BOTH x and y. If even one of them is negative, then |x|-|y| will never (x-y). It will be either -x-y (In case x is negative and y positive) OR x-(-y)-->x+y(x positive and y negative).

When y is negative, as |y| is always a non-negative entity, thus we can't write |y| = y. Thus, we attach a negative sign to 'y' to make the entire term (-y) as positive.

Also, even though (x+y) does look like addition of two numbers, it actually isn't, as y is negative.
It will be easier to understand this concept if you use valid nos.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  31 May 2013, 00:12
WholeLottaLove wrote:
I get the mechanics of flipping the signs when y is negative, but I guess I don't understand the logic.

If I take an absolute value of a number (always positive) then subtract from it an absolute value of a smaller number, which is negative how does that end up being X+Y? Are we looking just for the values of x and y, as opposed to the values of |x|-|y|?

----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.

x is greater than 0 => |x|=x
y is less than 0 => |y|=-y. You take -y from |y| because y<0
So |x|-|y|=x-(-y)=x+y

This is why |x|-|y| becomes x+y. If you want you can try with real numbers, example: x=5>0 and y=-3<0
|x|-|y|=|5|-|-3|=5-3=2
|x|-|y|=x+y=5-3=2
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  31 May 2013, 02:05
Expert's post
WholeLottaLove wrote:
How does x^2-2|xy|+y^2 become x^2+2xy+y^2? What happened to the -2|xy|? Even if xy were negative |xy| will be positive and pos. * neg (In this case, the -2) = negative, right?

Also, how do we know that xy is negative?

Step by step:

Step 1: square |x|-|y|=|x+y| --> (|x|-|y|)^2=(|x+y|)^2 (note that (|x+y|)^2=(x+y)^2) --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2

Step 2: cancel x^2+y^2 in both sides to get -2|xy|=2xy

Step 3: reduce by -2 to get |xy|=-xy

Step 4: apply absolute value property, which says that the absolute value is always non-negative --> LHS=|xy| is always non negative, thus |xy|\geq{0}, therefore RHS is also non-negative: |xy|=-xy\geq{0} --> -xy\geq{0}.

Step 5: multiply by -1 and flip the sign of the inequality --> xy\leq{0}

Step 6: apply info given in the stem --> since given that xy\neq{0}, then xy<0.

Hope it's clear.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  31 May 2013, 02:37
walker wrote:
By the way, I would be extremely careful with using the "squaring" approach to solve absolute value problems.

For example, if |x|=-|x|, what is x?

But,

(|x|)^2 = (-|x|)^2 --> x^2 = x^2 --> x can be any numbers (incorrect!).

|x|=-|x|

you shall write it as

2 |x| = 0 and then square it

you will get x=0

even without squaring you will get 0.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  10 Jul 2013, 09:10
If |x| - |y| = |x+y| and xy does not equal to 0, which of the following must be true?

|x| - |y| = |x+y|
|x|-|y| = an absolute value which means that |x|-|y| is positive. Therefore, we can square both sides.
(|x| - |y|)^2 = (|x+y|)^2
(|x| - |y|)*(|x| - |y|) = (|x+y|)*(|x+y|)
x^2-2(xy|) + y^2 = x^2+2xy+y^2
-2|xy|=2xy
(I take it we cannot add -2|xy| to 2xy?)
|xy|=-xy
-xy must be positive as it is equal to an absolute value. For that to be possible:
|xy|=-xy
|xy|=-(-xy)
|xy|=xy
xy=xy

So, xy < 0

(E)

A. x-y > 0
B. x-y < 0
C. x+y > 0
D. xy > 0
E. xy < 0
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  22 Jul 2013, 22:44
its better testing with some smart numbers.obviously will take some seconds more but worth it.
As per this question there can be four cases:
1 )+ +
2)+ -
3)- +
4)- -. Check for the validity and you will see that choices 2 and 3 only suffice.So the product is going to be <0 with either case.Hence E

Answer which you 'll derive using smart numbers will be beyond doubt and perfect.Get used to smart numbers in areas like absolute values,percentages,ratios.They help a lot )
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Re: PS - Number system [#permalink]  23 Apr 2014, 04:51
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}, but as given that xy\neq{0}, then xy<0.

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

Could you please explain following line as above defined in solution

note that (|x+y|)^2=(x+y)^2

Thanks.
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Re: PS - Number system [#permalink]  23 Apr 2014, 06:15
Expert's post
PathFinder007 wrote:
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}, but as given that xy\neq{0}, then xy<0.

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

Could you please explain following line as above defined in solution

note that (|x+y|)^2=(x+y)^2

Thanks.

Generally, |x|^2=x^2. For example, |-2|^2=4=2^2 or |3|^2=9=3^2.
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Re: PS - Number system [#permalink]  02 May 2014, 22:45
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}, but as given that xy\neq{0}, then xy<0.

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

HI Bunnel,

3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}

could you please clarify above statement. here y is positive so |x+y| should be -x+y

Thanks.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  02 May 2014, 23:05
divakarbio7 wrote:
If |x| - |y| = |x+y| and xy does not equal to 0, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y > 0
D. xy > 0
E. xy < 0

Let us plug in and eliminate:

x = -2 and y = 2 OR x = 2 and y = -2 then |2| - |-2| = |2-2|, let us check the options

A. ELIMINATED (-2 - 2>0)
B. ELIMINATED (2 - (-2) <0)
C. ELIMINATED (2 - 2 > 0)
D. ELIMINATED (-2(2) < 0 )
E. The only one which stays

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Re: PS - Number system [#permalink]  03 May 2014, 03:22
Expert's post
PathFinder007 wrote:
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}, but as given that xy\neq{0}, then xy<0.

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

HI Bunnel,

3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}

could you please clarify above statement. here y is positive so |x+y| should be -x+y

Thanks.

y is positive but x is negative and x is further from zero than y, so x+y is negative hence |x+y|=-(x+y). For example, x=-5 and y=1 --> x+y=-4=negative.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  06 May 2014, 20:00
My Analysis : xy # 0 mean x and y might have either positive or negative Numbers ( Integers)
Concepts : Absolute value , Modules
My Steps : As xy # 0 , So x and y have either positive or negative numbers
By Plugging x and y as positive numbers |x| - |y| = |x+y| the Equation never becomes Equal
So One Number has to be Negative
Ex : x = -6 , y = 3
|-6| - |3| = |-6 + 3|
6 - 3 = |-3 |
3 = 3
So xy = -6*3 = -18

Mean , xy < 0
E

If Any thing is Wrong Pls Reply
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Re: PS - Number system [#permalink]  18 Jun 2014, 05:57
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

Hi Bunuel ,

Can you please explain how |xy|=-xy --> xy\leq{0},
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Re: PS - Number system [#permalink]  18 Jun 2014, 07:13
Expert's post
gauravsoni wrote:
Bunuel wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

Hi Bunuel ,

Can you please explain how |xy|=-xy --> xy\leq{0},

Absolute value properties:

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.

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope this helps.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of [#permalink]  18 Jun 2014, 11:41
Bunuel wrote:
gauravsoni wrote:
divakarbio7 wrote:
if lxl - lyl = lx+yl anf xy does , not equal to o, which of the following must be true?

A. x-y > 0
B. x-y < 0
C. x+y >0
D. xy>0
E. xy<0

|x|-|y|=|x+y| --> square both sides --> (|x|-|y|)^2=(|x+y|)^2 --> note that (|x+y|)^2=(x+y)^2 --> (|x|-|y|)^2=(x+y)^2 --> x^2-2|xy|+y^2=x^2+2xy+y^2 --> |xy|=-xy --> xy\leq{0}

Another way:

Right hand side, |x+y|, is an absolute value, which is always non-negative, but as xy\neq{0}, then in this case it's positive --> RHS=|x+y|>0. So LHS must also be more than zero |x|-|y|>0, or |x|>|y|.

So we can have following 4 scenarios:
1. ------0--y----x--: 0<y<x --> |x|-|y|=x-y and |x+y|=x+y --> x-y\neq{x+y}. Not correct.
2. ----y--0------x--: y<0<x --> |x|-|y|=x+y and |x+y|=x+y --> x+y={x+y}. Correct.
3. --x------0--y----: x<0<y --> |x|-|y|=-x-y and |x+y|=-x-y --> -x-y={-x-y}. Correct.
4. --x----y--0------: x<y<0 --> |x|-|y|=-x+y and |x+y|=-x-y --> -x+y\neq{-x-y}. Not correct.

So we have that either y<0<x (case 2) or x<0<y (case 3) --> x and y have opposite signs --> xy<0.

Hope it helps.

Hi Bunuel ,

Can you please explain how |xy|=-xy --> xy\leq{0},

Absolute value properties:

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.

Theory on Abolute Values: math-absolute-value-modulus-86462.html

DS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=37
PS Abolute Values Questions to practice: search.php?search_id=tag&tag_id=58

Hard set on Abolute Values: inequality-and-absolute-value-questions-from-my-collection-86939.html

Hope this helps.[/quote]

Thanks a lot Bunuel.
Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of   [#permalink] 18 Jun 2014, 11:41
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