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

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If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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15 Oct 2014, 15:42
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Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0
[Reveal] Spoiler: OA

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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15 Oct 2014, 19:12
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Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

[Reveal] Spoiler:
Took me about 2 mins to complete. Feedback's are welcome, if I have missed or omitted anything from my method

Question stem mentions $$|X | + |Y| = -X -Y$$ --> from this you can drive that only possibility to satisfy the equation is when $$X & Y$$ both are negative.
Example: Assume $$X=-2$$ and $$Y=-1$$
Then, substitute in the given equation $$LHS = |-2|+|-1| = 2+1$$ & $$RHS = -(-2)-(-1) = 2+1 = 3$$
Therefore, $$LHS = RHS$$

Now, that you know $$X & Y$$ both are negative, substitute in each option-->

option A: $$X + Y > 0$$, therefore, $$-ve + -ve$$ will always be less than zero, hence does not satisfy the equation.
option B: $$X+Y < 0$$, therefore, $$-ve + -ve$$ will always be less than zero, hence satisfies the equation.
option C: $$X - Y > 0$$, therefore, $$(-ve) - (-ve)$$ can be less than zero or more than zero, hence does not satisfy the equation.
option D: $$X - Y < 0$$, therefore, $$(-ve) - (-ve)$$ can be less than zero or more than zero, hence does not satisfy the equation.
option E: $$X^2-Y^2 > 0$$, therefore, $$(-ve)^2 - (-ve)^2 = (+ve) - (+ve)$$ can be less than zero or more than zero, hence does not satisfy the equation.

Since only option B Satisfies the equation, the correct answer choice is option B

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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15 Oct 2014, 19:28
From given equation, it can be easily deduced that x<0, y<0
so (-ve) + (-ve) = (-ve) < 0

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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16 Oct 2014, 21:05
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Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

Good question: 20 sec solution

We know that for any $$|x|\geq{0}$$

We are told that $$xy \neq{0}$$that means neither x nor y is 0

Now in LHS we have |x|+|y|, which is greater than 0

So we have |x|+|y|> 0 or $$-x-y >0$$ or -(x+y)>0 or$$x+y<0$$

Ans B
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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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17 Oct 2014, 09:54
If x is <0, then |x| = -x ,
so From the given equation we can figure out that x and y are both negative numbers .
thus, -ve + (-Ve) will always be <0 hence B.

Bunuel please let me know if my approach is wrong !

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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18 Oct 2014, 03:15
according to question |x|+|y| = -x-y only when both x and y are negative(<0)..
The sum of two negative numbers is negative (<0)...hence B....

kudos:)

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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18 Oct 2014, 06:46
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Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

Given that, |x | + |y | = -x – y
LHS = |x | + |y |, implies that LHS > 0 {because mod is always +ve }

This implies that RHS > 0 i.e. -x – y > 0
- (X + Y ) > 0

Multiply by -1 on both sides and flip inequality sign

we get (X + Y ) < 0. Hence answer is B

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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01 Jan 2015, 14:07
WoundedTiger wrote:
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

Good question: 20 sec solution

We know that for any $$|x|\geq{0}$$

We are told that $$xy \neq{0}$$that means neither x nor y is 0

Now in LHS we have |x|+|y|, which is greater than 0

So we have |x|+|y|> 0 or $$-x-y >0$$ or -(x+y)>0 or$$x+y<0$$

Ans B

Isnt the rule |x|>0 when x is positive and |x|<0 when X is negative
(I know im mistaken, but I dont know where so please do point out )

so shouldn't -(x+y)<0

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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02 Jan 2015, 04:26
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Expert's post
WoundedTiger wrote:
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

Good question: 20 sec solution

We know that for any $$|x|\geq{0}$$

We are told that $$xy \neq{0}$$that means neither x nor y is 0

Now in LHS we have |x|+|y|, which is greater than 0

So we have |x|+|y|> 0 or $$-x-y >0$$ or -(x+y)>0 or$$x+y<0$$

Ans B

Isnt the rule |x|>0 when x is positive and |x|<0 when X is negative
(I know im mistaken, but I dont know where so please do point out )

so shouldn't -(x+y)<0

No. The absolute value of some expression is always non-negative: $$|some \ expression|\geq{0}$$, no matter whether that expression itself is positive, negative or 0.

When $$x \le 0$$ then $$|x|=-x$$, or more generally when $$\text{some expression} \le 0$$ then $$|\text{some expression}| = -(\text{some expression})$$. For example: $$|-5|=5=-(-5)$$;

When $$x \ge 0$$ then $$|x|=x$$, or more generally when $$\text{some expression} \ge 0$$ then $$|\text{some expression}| = \text{some expression}$$. For example: $$|5|=5$$.

Theory on Abolute Values: math-absolute-value-modulus-86462.html
Absolute value tips: absolute-value-tips-and-hints-175002.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 it helps.
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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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02 Jan 2015, 04:33
Much obliged brunel, makes much more sense.

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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25 May 2015, 07:22
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|x| + |y| = -x-y => - (x+y)

means that x+y<0 for their negative to be positive

B

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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26 May 2015, 20:59
Hi All,

This question can be solved by TESTing VALUES.

We're told that |X| + |Y| = -X -Y and that neither variable equals 0. We're asked which of the following answers MUST be true (which really means "which of the following is ALWAYS TRUE no matter how many different examples we come up with....?)

As complicated as this might look, it's really just telling us to use NEGATIVE values for X and Y. From the answer choices (and their focus on the relationship to 0), we should look to pick the SAME number for BOTH variables...

IF....
X = -1
Y = -1

Answer A: X+Y > 0 -1 -1 = -2 is NOT > 0. NOT the answer
Answer B: X+Y < 0 -1 - 1 = -2 IS < 0. This is a MATCH
Answer C: X-Y > 0 -1 +1 = 0 is NOT > 0. NOT the answer
Answer D: X-Y < 0 -1 +1 = 0 is NOT < 0. NOT the answer
Answer E: X^2 – Y^2 > 0 1 - 1 = 0 is NOT > 0. NOT the answer

[Reveal] Spoiler:
B

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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26 May 2015, 22:38
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Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

Going through the solutions posted above, I realized that while most students did answer the question right, some missed out on an important nuance:

It is incorrect to say that an absolute value expression is always positive.

The correct statement is:

An absolute value expression is always non-negative.

This means, |x| 0 (please note that I wrote ≥ here in this first expression, not just >)

Similarly, |y| 0

However,we are given here that xy doesn't equal 0. This means x ≠ 0 and y ≠ 0.

This is the reason why we can write |x| > 0
and |y| > 0

Adding the two inequalities, we get: |x| + |y| > 0

This means -x - y > 0 (since |x| + |y| = -x-y)
Or, -(x+y) > 0

That is, x+ y < 0

Here is a variation of the above question that highlights the importance of the green statement above:

If |x | + |y | = -x – y, which of the following can be true?

I. x + y < 0
II. x + y = 0
III. xy = 0

(A) I only
(B) III only
(C) I and II only
(D) II and III only
(E) I, II and III

Hope this helped!

Best Regards

Japinder
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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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07 May 2016, 09:27
bimalr9 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. X-Y < 0. E. X^2 - Y^2 > 0

This is how I solved. What is the correct way to do this problem?

[Reveal] Spoiler:
I assumed the only possible value of x and y to be negative as positive value will not satisfy the given information. Taking X as -1 and Y as -1. Now,
|-1|+|-1| = - (-1) + -(-1) == > 2 = 2.
Based on this I chose ‘B’ as answer. (–X) + (-Y) <0.

IMHO that's the perfect , shortest and the best approach to deal with the Particular problem, though there are numerous approaches for solving this problem as discussed above..

Answer will hence (B) be ( Same approach)
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If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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25 Jul 2016, 01:14
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

|x | + |y | = -x – y
Lets solve LHS
Anything that comes out of mod is positive
so |x|+|y| = postive
so |x|+|y| >0

Lets solve RHS
-x-y
-(x+y)

LHS =RHS
-(x+y)>0
-1(x+y)>0

Now since we know that our expression is positive we can remove the -1 by multiplying both side of the inequality with -1 and flipping the inequality sign (Don't forget whenever we multiply an equality with a negative number we have to flip the sign !!)
-1*-1(x+y)<0 * (-1)
1(x+y)<0
x+y<0

This is a correct match with our option B
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Re: If |x| + |y| = -x -y and xy does not equal 0, which of the following [#permalink]

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21 Sep 2016, 10:55
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achintsodhi wrote:
if |x| + |y| = -x -y and xy does not equal 0, which of the following must be true ?

a)$$x+y>0$$
b)$$x +y<0$$
c) $$x-y >0$$
d) $$x-y<0$$
e) $$x^2-Y^2>0$$

My second post ever...

|x| + |y| is always > 0;

Therefore -x - y > 0
-1*(x + y) > 0
x + y < 0

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Re: If |x| + |y| = -x -y and xy does not equal 0, which of the following [#permalink]

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21 Sep 2016, 11:22
|x| +|y| is always positive
Thus, -x -y is positive . So we can say that x and y are negative .

So sum of two negative number is negative

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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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15 May 2017, 23:04
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

We know that absolute values are always positive.

Therefore, |x | + |y | > 0. As |x | + |y | = -x – y, then -x – y > 0

-x – y= -(x+y) and -(x+y) > 0

To remove the negative sign, one must swap the inequality,

So (x+y) < 0

B
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Re: If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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16 May 2017, 08:05
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

I thought about the question as under:
LHS---> distance of x from zero + distance of y from zero i.e. something positive
RHS---> it MUST be positive, a quick glance over the options reveals B as the amswer....

Pl correct me if i have been plain lucky
kudos if i it helps

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If |x | + |y | = -x – y and xy does not equal 0, which of the followin [#permalink]

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08 Oct 2017, 23:50
Bunuel wrote:

Tough and Tricky questions: Absolute Values.

If |x | + |y | = -x – y and xy does not equal 0, which of the following must be true?

A. x + y > 0
B. x + y < 0
C. x – y > 0
D. x – y < 0
E. x^2 – y^2 > 0

When dealing with this question, it maybe solved with logic or plug-in numbers or both.

$$xy\neq{0}$$......$$x\neq{0}$$ & $$y\neq{0}$$

Analyzing RHS:

$$|x | + |y |$$ = |any number with any sign | + |any number with any sign| = POSITIVE Number

Analyzing LHS:

$$-x – y = - (x+y)$$ .......MUST BE positive to match RHS

$$- (x+y) > 0$$....then $$x+y<0$$

Another approach

Plug-in numbers. Here sense of numbers says 2 notes:

1- It useless to use positive examples

2- No restriction if x=y. This note will help to DISPROVE the choices as all related to zero

3- In absolute questions, I usually prefer to start with negative values

Let $$x = y = -2$$

$$x -y = 0$$

x - y = 0.........It means choices C, D & E are disproved........Eliminate

x + y = -4

Choice A is disapproved..................Eliminate

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If |x | + |y | = -x – y and xy does not equal 0, which of the followin   [#permalink] 08 Oct 2017, 23:50

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