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Retired Moderator B
Joined: 05 Jul 2006
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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Interpreting the given

the distance between x and 0 is larger than that between y and 0 and this difference in distance of each of x and y from zero is equal to the distance between x and -y

draw this on a number line

..................x..............-y...............0.............y

this means that x and y has to be on opposite sides from zero , i.e. different signs thus xy<0
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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can we do the following

Generally /x/ - /-y/ <= /x-(-y)/ and the equality hold true only when -xy>o or xy<0 and /x/>/-y/ or /x/>/y/

given xy not = 0 , and given /x/-/y/ = /x+y/ , i.e. /x/ - /-y/ = /x-(-y)/ therefore /x/> /-y/ and -xy>o ,i.e xy<0
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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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?

Originally posted by WholeLottaLove on 30 May 2013, 15:25.
Last edited by WholeLottaLove on 30 May 2013, 15:46, edited 1 time in total.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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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?

It does not become xy.

$$|x| - |y| = |x+y|$$ , square both sides $$(|x| - |y|)^2 = (|x+y|)^2$$, $$x^2 - 2|xy|=y^2 = x^2+2xy+y^2$$, eliminate similar terms and divide by 2
$$-|xy|=xy$$ from here multiply by -1 just for clarity and obtain $$|xy|=-xy$$
remember that $$|abs|\geq{0}$$ to the equation translates into $$-xy\geq{0}$$ or $$xy\leq{0}$$

The text says that $$xy\neq{0}$$ so $$xy\leq{0}$$=>$$xy<0$$
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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but I don't understand how the -2|xy| simplifies to + 2xy.

And why do we multiply by -1?!

Zarrolou wrote:
$$|x| - |y| = |x+y|$$ , square both sides $$(|x| - |y|)^2 = (|x+y|)^2$$, $$x^2 - 2|xy|=y^2 = x^2+2xy+y^2$$, eliminate similar terms and divide by 2
$$-|xy|=xy$$ from here multiply by -1 just for clarity and obtain $$|xy|=-xy$$
remember that $$|abs|\geq{0}$$ to the equation translates into $$-xy\geq{0}$$ or $$xy\leq{0}$$

The text says that $$xy\neq{0}$$ so $$xy\leq{0}$$=>$$xy<0$$
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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WholeLottaLove wrote:
but I don't understand how the -2|xy| simplifies to + 2xy.

And why do we multiply by -1?!

You arrive at $$-2|xy|=2xy$$ right? Now divide by 2: $$-|xy|=xy$$.
From here the -1 multiplication is not necessary to arrive at the result, but IMO it helps.

Keeping in mind that $$|abs|\geq{0}$$, we can write $$-|abs|\leq{0}$$, right?
The abs expression stands for any value or expression we find inside "| |", xy included.

Now we can merge the equations $$xy=-|xy|$$ with $$-|abs|\leq{0}$$ into $$xy=-|xy|\leq{0}$$ => $$xy\leq{0}$$
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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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?

I am sorry about the stupid questions. This is an extremely difficult topic for me and I am having a tough time picking it up.

Zarrolou wrote:
WholeLottaLove wrote:
but I don't understand how the -2|xy| simplifies to + 2xy.

And why do we multiply by -1?!

You arrive at $$-2|xy|=2xy$$ right? Now divide by 2: $$-|xy|=xy$$.
From here the -1 multiplication is not necessary to arrive at the result, but IMO it helps.

Keeping in mind that $$|abs|\geq{0}$$, we can write $$-|abs|\leq{0}$$, right?
The abs expression stands for any value or expression we find inside "| |", xy included.

Now we can merge the equations $$xy=-|xy|$$ with $$-|abs|\leq{0}$$ into $$xy=-|xy|\leq{0}$$ => $$xy\leq{0}$$
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GMAT 1: 750 Q50 V40 Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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2
Another 15 sec approach:

1. The equation isn't sensitive to changing signs of x and y simultaneously. In other words, the equation is the same for (-x,-y). So, A, B, C are out.
2. If x and y had the same sign, |x+y| would be always greater than |x| (and |x| - |y|). So, x and y have different signs and only E remains.
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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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?

I am sorry about the stupid questions. This is an extremely difficult topic for me and I am having a tough time picking it up.

The text says so.

You start from here $$|x| - |y| = |x+y|$$, square both side ...(refer to the passages above)
Eventually you obtain $$-|xy|=xy$$

In the Bunuel's example:
2. ----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 => $$|y|=-y$$
So $$|x|-|y|=x-(-y)=x+y$$

But those two examples are from two different approaches, do not mix them
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GMAT 1: 750 Q50 V40 Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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1
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!).
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Re: If |x| - |y| = |x+y| and xy does not equal to 0, which of  [#permalink]

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1
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]

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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]

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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]

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7
9
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]

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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]

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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]

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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]

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1
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]

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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]

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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. Re: PS - Number system   [#permalink] 02 May 2014, 23:45

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# If |x| - |y| = |x+y| and xy not equal zero , which of the following  