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GMAT 1: 680 Q48 V34 If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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39 00:00

Difficulty:   75% (hard)

Question Stats: 63% (02:37) correct 37% (02:55) wrong based on 589 sessions

### HideShow timer Statistics If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

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Originally posted by Hussain15 on 13 Jun 2010, 05:41.
Last edited by Bunuel on 14 Aug 2012, 00:31, edited 1 time in total.
Edited the question.
Math Expert V
Joined: 02 Sep 2009
Posts: 56266
Re: Interesting Absolute value Problem  [#permalink]

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13
6
Hussain15 wrote:
If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

If $$x\leq{0}$$, then $$x + |x| + y = 7$$ becomes: $$x-x+y=7$$ --> $$y=7>0$$, but then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, which contradicts the initial assumption $$x\leq{0}$$. So $$x$$ can not be $$\leq{0}$$ --> hence $$x>0$$.

Similarly if $$y\geq{0}$$, then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, but then $$x + |x| + y = 7$$ becomes: $$x+x+y=12+y=7$$ --> $$y=-5<0$$, which contradicts the initial assumption $$y\geq{0}$$. So $$y$$ can not be $$\geq{0}$$ --> hence $$y<0$$.

So $$x>0$$ and $$y<0$$:
$$x+|x|+y=7$$ becomes: $$x+x+y=7$$ --> $$2x+y=7$$;
$$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Solving: $$x=4$$ and $$y=-1$$ --> $$x+y=3$$.

I feel there is an easier way, but world cup makes it harder to concentrate.
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Re: Interesting Absolute value Problem  [#permalink]

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4
1
I did it by brute force, considering all possibilities. But I'm sure someone can come up with a way to quickly identify the signs of x and y.

x + |x| + y = 7 ....1

x + |y| - y = 6 .....2

Consider x < 0
=> |x| = -x
Substitute in 1

x - x + y = 7
y = 7

Substitute in 2

x + 7 - 7 = 6
x = 6

But this violates x < 0 So our assumption was incorrect.

Consider x > 0 and y > 0
=> |x| = x and |y| = y

Substitute in 2

x + y - y = 6
x = 6

Substitute in 1

6 + 6 + y = 7
y = -5

This violates y > 0 so our assumption was incorrect.

Consider x > 0 and y < 0
=> |x| = x and |y| = -y

Substituting in 1 and 2:

2x + y = 7 .....3
x - 2y = 6 .....4

Solving we get x = 4 and y = -1
x + y = 3

Pick A.
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Re: Interesting Absolute value Problem  [#permalink]

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1
Bunuel wrote:
Hussain15 wrote:
If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

If $$x\leq{0}$$, then $$x + |x| + y = 7$$ becomes: $$x-x+y=7$$ --> $$y=7>0$$, but then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, which contradicts the initial assumption $$x\leq{0}$$. So $$x$$ can not be $$\leq{0}$$ --> hence $$x>0$$.

Similarly if $$y\geq{0}$$, then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, but then $$x + |x| + y = 7$$ becomes: $$x+x+y=12+y=7$$ --> $$y=-5<0$$, which contradicts the initial assumption $$y\geq{0}$$. So $$y$$ can not be $$\geq{0}$$ --> hence $$y<0$$.

So $$x>0$$ and $$y<0$$:
$$x+|x|+y=7$$ becomes: $$x-x+y=7$$ --> $$2x+y=7$$;
$$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Solving: $$x=4$$ and $$y=-1$$ --> $$x+y=3$$.

I feel there is an easier way, but world cup makes it harder to concentrate.

Why when y<0 do we get -2y?
Math Expert V
Joined: 02 Sep 2009
Posts: 56266
Re: Interesting Absolute value Problem  [#permalink]

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GMATBLACKBELT720 wrote:
Bunuel wrote:
Hussain15 wrote:
If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

If $$x\leq{0}$$, then $$x + |x| + y = 7$$ becomes: $$x-x+y=7$$ --> $$y=7>0$$, but then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, which contradicts the initial assumption $$x\leq{0}$$. So $$x$$ can not be $$\leq{0}$$ --> hence $$x>0$$.

Similarly if $$y\geq{0}$$, then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, but then $$x + |x| + y = 7$$ becomes: $$x+x+y=12+y=7$$ --> $$y=-5<0$$, which contradicts the initial assumption $$y\geq{0}$$. So $$y$$ can not be $$\geq{0}$$ --> hence $$y<0$$.

So $$x>0$$ and $$y<0$$:
$$x+|x|+y=7$$ becomes: $$x-x+y=7$$ --> $$2x+y=7$$;
$$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Solving: $$x=4$$ and $$y=-1$$ --> $$x+y=3$$.

I feel there is an easier way, but world cup makes it harder to concentrate.

Why when y<0 do we get -2y?

When $$y<0$$, then $$|y|=-y$$ and $$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.
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Re: Interesting Absolute value Problem  [#permalink]

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^
Wow... confusing as heck. Essentially saying that (hypothetical number here) |-3| = -(-3), thats fine. But I kept thinking you would apply this to -y and essentially make it +y since and make them both +y...
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Re: Interesting Absolute value Problem  [#permalink]

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1
for x>0 and x < 0
2x+ y = 7 and y = 7

for y>0 and y<0
x=6 and x-2y=6

for x,y<0 solution exists.
solving x=4 and y = -1

3
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

can be done in 2.3 mins :

there are 4 cases to be tested :
1) x is -ve and y is -ve
substituting in the equation , we get x-x+y=7 and x-y-y=6 solve for x and y we get x=20 and y=7 , so x+y=27 REJECT

2)x is +ve and y is +ve

substitute in the equation, we ger x+x+y=7 and x+y-y=6 solve for x and y we get x=6 and y=-5 ,therefore x+y=1 not on list so REJECT

3) x is -ve and y is +ve

substitute , we get x-x=y=7 and x+y-y=6 solve fo x and y we get x=6 and y=7, x+y=13 not on list so REJECT

4) x is +ve and y is -ve

substitute , we get x+x=y=7 and x-y-y=6 solve for x and y , we get x=4 and y= -1 ,x+y=3 , ANSWER CHOICE
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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I solved in 1.36 mins _________________
Kudos always encourages me
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Posts: 44
Re: Interesting Absolute value Problem  [#permalink]

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Bunuel wrote:
GMATBLACKBELT720 wrote:
Bunuel wrote:

If $$x\leq{0}$$, then $$x + |x| + y = 7$$ becomes: $$x-x+y=7$$ --> $$y=7>0$$, but then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, which contradicts the initial assumption $$x\leq{0}$$. So $$x$$ can not be $$\leq{0}$$ --> hence $$x>0$$.

Similarly if $$y\geq{0}$$, then $$x + |y| - y =6$$ becomes: $$x+y-y=6$$ --> $$x=6>0$$, but then $$x + |x| + y = 7$$ becomes: $$x+x+y=12+y=7$$ --> $$y=-5<0$$, which contradicts the initial assumption $$y\geq{0}$$. So $$y$$ can not be $$\geq{0}$$ --> hence $$y<0$$.

So $$x>0$$ and $$y<0$$:
$$x+|x|+y=7$$ becomes: $$x-x+y=7$$ --> $$2x+y=7$$;
$$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Solving: $$x=4$$ and $$y=-1$$ --> $$x+y=3$$.

I feel there is an easier way, but world cup makes it harder to concentrate.

Why when y<0 do we get -2y?

When $$y<0$$, then $$|y|=-y$$ and $$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Sorry, still confusing me.
I understand the first y i.e.
|y| = -y
But y < 0,
so wouldn't x + |y| - y = x - y - (-y), which would make it just x?
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Posts: 56266
Re: Interesting Absolute value Problem  [#permalink]

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jjack0310 wrote:
Bunuel wrote:
GMATBLACKBELT720 wrote:

Why when y<0 do we get -2y?

When $$y<0$$, then $$|y|=-y$$ and $$x+|y|-y=6$$ becomes: $$x-y-y=6$$ --> $$x-2y=6$$.

Sorry, still confusing me.
I understand the first y i.e.
|y| = -y
But y < 0,
so wouldn't x + |y| - y = x - y - (-y), which would make it just x?

When y<0, then |y|=-y: correct. But y must stay as it is.

Consider this, suppose we have only x-y=6, and I tell you that y is negative would you rewrite the equation as x-(-y)=6?

Hope it's clear.
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

x>0, y>0

x + (x) + y = 7
2x+y=7

x + |y| - y = 6
x + y - y = 6
x=6

2x+y=7
2(6)+y=7
y=-5

INVALID as 6>0 but -5 is not > 0

x>0, y<0

x + (x) + y = 7
2x+y=7

x + |y| - y = 6
x + (-y) - y = 6
x - 2y = 6
x = 6+2y

2(6+2y) + y = 7
12+4y+y=7
12+5y=7
5y+-5
y=-1

2x+y=7
2x + (-1) = 7
2x - 1 = 7
2x=8
x=4

VALID as 4>0 and -1<0

therefore;

x+y = (4)+(-1) = 3

(A)
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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1
Hussain15 wrote:
If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

In such a question, you can use some brute force and get to the answer too. How long it takes depends on how quickly you observe the little things.

x + |x| + y = 7
x + |y| - y =6

Both equations yield about the same result though in one y is positive and in the other it is negative. |x| and |y| are positive and assuming x is positive, a negative y would pull down the first equation and pump up the second one to give almost equal values. The difference is very small also signifies that the negative variable might have a very small value.
Since the options give the value of x + y as 3/4/5... etc, it is likely that we are dealing with small number pairs such as (4, 2), (3, 2), (4, 1) etc.
Since the first equation has 7 as the result, both variables will not be even.
A couple of quick iterations brought me to (4, -1).
So x + y = 3
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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VeritasPrepKarishma wrote:
Hussain15 wrote:
If x + |x| + y = 7 and x + |y| - y =6 , then x + y =

A. 3
B. 4
C. 5
D. 6
E. 9

In such a question, you can use some brute force and get to the answer too. How long it takes depends on how quickly you observe the little things.

x + |x| + y = 7
x + |y| - y =6

Both equations yield about the same result though in one y is positive and in the other it is negative. |x| and |y| are positive and assuming x is positive, a negative y would pull down the first equation and pump up the second one to give almost equal values. The difference is very small also signifies that the negative variable might have a very small value.
Since the options give the value of x + y as 3/4/5... etc, it is likely that we are dealing with small number pairs such as (4, 2), (3, 2), (4, 1) etc.
Since the first equation has 7 as the result, both variables will not be even.
A couple of quick iterations brought me to (4, -1).
So x + y = 3

Hi Karishma,

I tried another approach, but got stuck - can you hep me solve by this method?
x + |x| + y = 7 => x + y = 7 - |x| => Whatever be the value of x, |x| will be always non-negative => x + y has to be less than 7 => This eliminates option E
x + |y| - y = 6 => Now, if y is positive, then this equation becomes x = 6 => on substituting in above eqn, we get y = -5 => x+y = 1 => Not present in any of the options => y is negative => x - y - y = 6 => x - 2y = 6 => x = 6 + 2y
Now x + y = 7 - |6 + 2y|
If we take y = -1, we get x + y = 3 = Option A
If we take y = -2, we get x + y = 5 = Option C
I am getting both options here - where am I going wrong here? Or this approach incorrect?
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If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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2
sagar2911 wrote:

Hi Karishma,

I tried another approach, but got stuck - can you hep me solve by this method?
x + |x| + y = 7 => x + y = 7 - |x| => Whatever be the value of x, |x| will be always non-negative => x + y has to be less than 7 => This eliminates option E
x + |y| - y = 6 => Now, if y is positive, then this equation becomes x = 6 => on substituting in above eqn, we get y = -5 => x+y = 1 => Not present in any of the options => y is negative => x - y - y = 6 => x - 2y = 6 => x = 6 + 2y
Now x + y = 7 - |6 + 2y|
If we take y = -1, we get x + y = 3 = Option A
If we take y = -2, we get x + y = 5 = Option C
I am getting both options here - where am I going wrong here? Or this approach incorrect?

You are making a mistake with using |6+2y|

After you get x=6+2y and substitute in equation (1), you get 6+2y+|6+2y|+y=7 ---->6+3y+|6+2y| = 7 ---> 3y+|6+2y| = 1. Now the point to note is that the 'nature' of |6+2y| changes at y=-3 and thus you need to evaluate |6+2y| for values smaller than -3 and for values greater than -3.

You have already assumed that y<0 in order to get x=6+2y, so your ranges to consider become -3 $$\le$$q y < 0 and y < -3

Case 1: -3 $$\le$$ y < 0, giving you |6+2y| $$\geq$$ 0 ----> 3y+|6+2y| = 1 ---> 3y+6+2y = 1 ---> 5y=-5 ---> y=-1. Acceptable value giving you x=6+2y = 4 --> x+y = 3

Case 2: y<-3 ---> |6+2y| = -(6+2y) ---> 3y+|6+2y| = 1 ---> 3y-6-2y = 1 ---> y=7 this contradicts the assumption that y<-3 , making this out of scope.

Thus the only value of x+y = 3.
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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Engr2012 wrote:
sagar2911 wrote:

Hi Karishma,

I tried another approach, but got stuck - can you hep me solve by this method?
x + |x| + y = 7 => x + y = 7 - |x| => Whatever be the value of x, |x| will be always non-negative => x + y has to be less than 7 => This eliminates option E
x + |y| - y = 6 => Now, if y is positive, then this equation becomes x = 6 => on substituting in above eqn, we get y = -5 => x+y = 1 => Not present in any of the options => y is negative => x - y - y = 6 => x - 2y = 6 => x = 6 + 2y
Now x + y = 7 - |6 + 2y|
If we take y = -1, we get x + y = 3 = Option A
If we take y = -2, we get x + y = 5 = Option C
I am getting both options here - where am I going wrong here? Or this approach incorrect?

You are making a mistake with using |6+2y|

After you get x=6+2y and substitute in equation (1), you get 6+2y+|6+2y|+y=7 ---->6+3y+|6+2y| = 7 ---> 3y+|6+2y| = 1. Now the point to note is that the 'nature' of |6+2y| changes at y=-3 and thus you need to evaluate |6+2y| for values smaller than -3 and for values greater than -3.

You have already assumed that y<0 in order to get x=6+2y, so your ranges to consider become -3 $$\le$$q y < 0 and y < -3

Case 1: -3 $$\le$$ y < 0, giving you |6+2y| $$\geq$$ 0 ----> 3y+|6+2y| = 1 ---> 3y+6+2y = 1 ---> 5y=-5 ---> y=-1. Acceptable value giving you x=6+2y = 4 --> x+y = 3

Case 2: y<-3 ---> |6+2y| = -(6+2y) ---> 3y+|6+2y| = 1 ---> 3y-6-2y = 1 ---> y=7 this contradicts the assumption that y<-3 , making this out of scope.

Thus the only value of x+y = 3.

Excellent. Thank you dude! _________________
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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I understand that if you first that x= pos. and y= neg. you get x=6 and y=-5.

But then this wouldn't be right, because it would contradict the assumptions that x should be neg. and y should be pos.

I don't understand, where do i get these assumptions from?
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Re: If x + |x| + y = 7 and x + |y| - y =6 , then x + y =  [#permalink]

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