If x+|x|+y=7 and x+|y|-y=6 what is x+y=? : GMAT Problem Solving (PS)
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# If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

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If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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18 Feb 2013, 05:54
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If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13
[Reveal] Spoiler: OA

Last edited by Bunuel on 18 Feb 2013, 05:57, edited 1 time in total.
Renamed the topic and edited the question.
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guerrero25 wrote:
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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18 Mar 2013, 12:44
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The first thing you should keep in mind when facing modulus (absolute value) problems is the definition of the modulus :

- If x > 0 then |x| = x > 0 (1) ;
- If x < 0 then |x| = - x > 0 (2) ;
-|x| IS ALWAYS POSITIVE NO MATTER x ! ;

Now let's solve the problem : (It's a pretty lengthy solution but I'd rather not miss anything )

We need to find the value of x+y knowing that : x+|x|+y=7 and x+|y|-y=6

Now considering the definition of the modulus, we have 4 cases :

- Case 1 : x > 0 and y > 0

According to (1) we'll have :

x+|x|+y = 7 => x+x+y = 7 => 2x + y = 7 => 12 + y = 7 => y = -5 (which is contradictory with the fact that y is positive)
x+|y|-y = 6 => x+y-y = 6 => x = 6

- Case 2 : x > 0 and y < 0

According to (1) and (2) we'll have :

x+|x|+y = 7 => x+x+y = 7 => 2x + y = 7 (E1)
x+|y|-y = 6 => x-y-y = 6 => x - 2y = 6 (E2)

Since we have two equations, we can manipulate them to find the value of x and y separately. In this case, multiplying both (E1) and (E2) by 2 then substracting each other will yield : x = 4 and y = - 1 (which is consistent with the fact that x is positive and y is negative). So x+y = 3 which is answer choice C.

But we're not stopping here

We have to make sure that the other cases don't contradict what we've found.

- Case 3 : x < 0 and y > 0

According to (1) and (2) we'll have :

x+|x|+y = 7 => x-x+y = 7 => y = 7
x+|y|-y = 6 => x+y-y = 6 => x = 6 (which is contradictory with the fact that x is negative)

- Case 4 : x < 0 and y < 0

According to (2) we'll have :

x+|x|+y = 7 => x-x+y = 7 => y = 7 (which is contradictory with the fact that y is negative)
x+|y|-y = 6 => x-y-y = 6 => x - 2y = 6

So all in all, only case 2 was valid in this case and the correct answer is C.

Note : The method may seem lengthy and daunting but when you know the modulus definition and are comfortable manipulating variables, it should be a breeze.

Hope that helped :D
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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18 Feb 2013, 19:57
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guerrero25 wrote:
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

You can also solve this question using some brute force.

Notice x+|y|-y=6
The first thing that comes to mind is that if y is positive or 0, x = 6. But when we put x = 6 in x+|x|+y=7, we get y negative. So y cannot be positive or 0. y must be negative. So, x+|y|-y=6 becomes x - 2y = 6

Since most options are positive values, it is very likely that x is positive so x+|x|+y=7 becomes 2x+y = 7. You can obtain 7 by subtracting 1 from 8.
If x = 4 and y = -1, both equations are satisfied.
So x+y = 4 - 1 = 3

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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 24 Jan 2013 Posts: 9 Followers: 0 Kudos [?]: 9 [6] , given: 5 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 23 Feb 2013, 23:59 6 This post received KUDOS 3 This post was BOOKMARKED There can be only four different cases: Case 1: x<0 , y>0 If this is true then |x| = -x & |y|= y Equation would bcom : x-x+y = 7 & x+y-y=6 , which means y=7 & x= 6 which can not be true as we assumed x<0, thus rejected Case 2 : x<0 , y<0 If this holds trues then |x| = -x & |y|=- y Equation would bcom : x-x+y = 7 & x-y-y=6 , which means y=7 & x= 20 which can not be true as we assumed x<0,y<0 thus rejected CASE 3 : x>0 , y >0 If this holds trues then |x| = x & |y|= y Equation would bcom : x+x+y = 7 & x+y-y=6 , which means y=-5 & x= 6 which can not be true as we assumed ,y>0 thus rejected Case 4 : x>0, y <0 If this holds trues then |x| = x & |y|=- y Equation would bcom : x+x+y = 7 & x-y-y=6 , solving these equations we get y=-1 & x= 4 which can be true as it meets our assumption of x>0 & y<0, thus accepted therefore answer x+y= 4-1= 3 hence C. Intern Status: Currently Preparing the GMAT Joined: 15 Feb 2013 Posts: 31 Location: United States GMAT 1: 550 Q47 V23 GPA: 3.7 WE: Analyst (Consulting) Followers: 1 Kudos [?]: 17 [3] , given: 11 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 19 Mar 2013, 08:21 3 This post received KUDOS Reevak wrote: Virgilius wrote: Reevak, the algebraic definition of the modulus is : - If x > 0 then |x| = x > 0 ; - If x < 0 then |x| = -x < 0 ; VeritasPrepKarishma wrote: |y| = |-3| = 3 This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign. Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative) Thanks a lot guys! Now there is one thing I still not understand, and another that I do understand. Now I see that if y = - 3 then |-3| = 3 ≠ y Instead |-3| = 3 = - y (since the minus sign will turn y positive turning the equality true) This I understand. However, there is still one thing I don't get. If y < 0 then shouldn't this equation x + |y| - y = 6 be equal to x - y + y = 6? Instead, you mentioned it should be x - y - y = 6 but I don't understand why Why - y and not + y? Is -(-3) not 3? You don't have to change the "-" sign to a "+" sign, since that sign is proper to the equation and not the variable. Let's assume that y = - 3 (y is negative) and x = 0 If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false. My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!! Hope that was clear enough. VP Status: Final Lap Up!!! Affiliations: NYK Line Joined: 21 Sep 2012 Posts: 1096 Location: India GMAT 1: 410 Q35 V11 GMAT 2: 530 Q44 V20 GMAT 3: 630 Q45 V31 GPA: 3.84 WE: Engineering (Transportation) Followers: 38 Kudos [?]: 532 [2] , given: 70 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 18 Feb 2013, 07:28 2 This post received KUDOS Bunuel wrote: guerrero25 wrote: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? A. 1 B. -1 C. 3 D. 5 E. 13 If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0. If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases). Answer: C. Hi Bunuel Adding both eqn we get 2x +IxI+IyI = 13 Now considering X <0 and y>0 2x-x +y = 13 we get x+y =13 Hence answer should be E Correct me if i am wrong!!! Archit Math Expert Joined: 02 Sep 2009 Posts: 37102 Followers: 7251 Kudos [?]: 96457 [2] , given: 10751 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 18 Feb 2013, 07:37 2 This post received KUDOS Expert's post Archit143 wrote: Bunuel wrote: guerrero25 wrote: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? A. 1 B. -1 C. 3 D. 5 E. 13 If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0. If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases). Answer: C. Hi Bunuel Adding both eqn we get 2x +IxI+IyI = 13 Now considering X <0 and y>0 2x-x +y = 13 we get x+y =13 Hence answer should be E Correct me if i am wrong!!! Archit OA is C, not E, so yes your solution is not correct. If x<0 and y>0, then we'll have x-x+y=7 and x+y-y=6. From the second equation x=6, so we can discard this case as y here is not more than 0. _________________ Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7185 Location: Pune, India Followers: 2167 Kudos [?]: 14019 [2] , given: 222 Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink] ### Show Tags 18 Mar 2013, 19:44 2 This post received KUDOS Expert's post 1 This post was BOOKMARKED Reevak wrote: VeritasPrepKarishma wrote: y must be negative. So, x+|y|-y=6 becomes x - 2y = 6 I don't understand the reasoning behind this statement. If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6 The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition. What am I missing? Say y = -3, what is the value of |y|? Substitute -3 in place of y |y| = |-3| = 3 This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign. Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative) We define mod as: |y| = y if y is positive |y| = -y if y is negative On the other hand, if I ask you: what is the value of y? It is -3 only What is the value of -y? It is -(-3) = 3 x+|y|-y=6 If we know that y is negative, |y| = -y (which becomes positive) x - y - y = 6 (notice that the -y in the original equation stays the same since it has no mod around it) _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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18 Feb 2013, 06:22
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guerrero25 wrote:
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

for x>0 ; y>0 => x=6, y=-5 ------ [1]
for x>0 ; y<0 => x=4, y=-1 => x+y = 3 -------- [2]
for x<0 ; y>0 => x =6, y=7 ----------- [3]
for x<0, y<0 => y=7, x =20 ----------- [4]

Hence, x+y = 3
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18 Feb 2013, 08:23
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Hi Bunuel
Is there any specific reason for applying x<0 and y>0 separately to the equation. I mean the signs could have been applied to the sum of both equations.
Another question , why did you not apply the both the signs to one equation only...

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18 Mar 2013, 11:28
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VeritasPrepKarishma wrote:
y must be negative. So, x+|y|-y=6 becomes x - 2y = 6

I don't understand the reasoning behind this statement.

If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6

The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition.

What am I missing?
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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18 Mar 2013, 12:24
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Reevak wrote:
VeritasPrepKarishma wrote:
y must be negative. So, x+|y|-y=6 becomes x - 2y = 6

I don't understand the reasoning behind this statement.

If y is negative, then x+|y|-y=6 becomes x +|-y|-(-y)=6 therefore x + y + y = 6 finally becomes x +2y=6

The absolute value of a negative integer is a positive value, and the subtraction of a negative integer results in an addition.

What am I missing?

Reevak, the algebraic definition of the modulus is :

- If x > 0 then |x| = x > 0 ; (1)
- If x < 0 then |x| = -x > 0 ; (2)

The reason why your statement is incorrect is because you altered the equation.

You can't put a "-" sign to a variable to say that it's negative. In fact, in our case, since y is negative, putting a "-" sign to it will make it positive, thus altering your equation and guiding you to a wrong answer.

What you did is the same as doing the following :

y = - 3 is negative. Therefore - y (which is still negative according to your syntax) will be equal to -y = -(-3) = 3 which is NOT negative.

So be careful with your computations.

Now let's get back to your issue :

Since y is negative ( y < 0 ), then according to (2) we'll have |y| = - y.

Injecting it into our equation yields : x+|y|-y = 6 => x - y - y = 6 => x - 2y =6. Hence the result.

You should be fine if you remember the definition of the modulus. I know it can be tricky at first, but once you get used to it, it'll become second-nature to you

Hope that helped.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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19 Mar 2013, 08:05
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Virgilius wrote:
Reevak, the algebraic definition of the modulus is :

- If x > 0 then |x| = x > 0 ;
- If x < 0 then |x| = -x < 0 ;

VeritasPrepKarishma wrote:
|y| = |-3| = 3
This is what you mean by mod of a negative number is positive. The number becomes positive when you remove the mod sign.
Notice that when y is negative, |y| is not equal to y. It is equal to -y (which, by the way, is a positive number since y is negative)

Thanks a lot guys! Now there is one thing I still not understand, and another that I do understand.

Now I see that if y = - 3 then |-3| = 3 ≠ y
Instead |-3| = 3 = - y (since the minus sign will turn y positive turning the equality true)
This I understand.

However, there is still one thing I don't get.
If y < 0 then shouldn't this equation x + |y| - y = 6 be equal to x - y + y = 6?

Instead, you mentioned it should be x - y - y = 6 but I don't understand why
Why - y and not + y?
Is -(-3) not 3?
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19 Mar 2013, 09:28
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Virgilius wrote:
If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false.

My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!!

Hope that was clear enough.

Yes, now it's clear!
I was changing the sign of the equation and by doing that I was changing the sign of the variable.
Now I see my mistake.

Thanks Virgilius, you were of great help.

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19 Mar 2013, 09:31
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Reevak wrote:
Virgilius wrote:
If what you said was true, meaning that if y is negative, x + |y| - y = 6 will be equal to x - y + y = 6, we'll have for this example : 0 - (-3) + (-3) = 0 + 3 - 3 = 0 ≠ 6, which is obviously false.

My point is : the modulus is the only fonction that can change the sign of a variable or an expression based on its related sign (whether the variable or expression is positive/negative). Don't change the signs that are proper to the equation itself !!!

Hope that was clear enough.

Yes, now it's clear!
I was changing the sign of the equation and by doing that I was changing the sign of the variable.
Now I see my mistake.

Thanks Virgilius, you were of great help.

Kudos

No worries

It's important for anyone prepping the GMAT to avoid these little confusions. I was a little apprehensive that my explanation wasn't clear enough for you. But I'm glad it was in the end.

Good luck.
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=? [#permalink]

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25 Apr 2013, 22:53
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Bunuel wrote:
guerrero25 wrote:
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Hi Bunuel,
I didn't understand how the second equation became x-2y=6. If y<0, won't it be x+l-yl-(-y)=6 => x+2y=6 ?
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25 Apr 2013, 23:51
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Expert's post
sharmila79 wrote:
Bunuel wrote:
guerrero25 wrote:
If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

A. 1
B. -1
C. 3
D. 5
E. 13

If x<0 and y<0, then we'll have x-x+y=7 and x-y-y=6. From the first equation y=7, so we can discard this case since y is not less than 0.

If x>=0 and y<0, then we'll have x+x+y=7 and x-y-y=6. Solving gives x=4>0 and y=-1<0 --> x+y=3. Since in PS questions only one answer choice can be correct, then the answer is C (so, we can stop here and not even consider other two cases).

Hi Bunuel,
I didn't understand how the second equation became x-2y=6. If y<0, won't it be x+l-yl-(-y)=6 => x+2y=6 ?

No! When $$y<0$$, then $$|y|=-y$$, thus $$x+|y|-y=6$$ becomes $$x-y-y=6$$ --> $$x-2y=6$$.
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14 Jun 2013, 07:30
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WholeLottaLove wrote:
Ok, so here is my question:

Let's say we look for all the cases of x,y being positive and negative (four in total)

(x is negative)

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

If x is negative, why don't we plug in -x for all values of x in both equations (i.e. x+|x|+y=7 ==> -x+-x+y=7)

That is not a valid solution:

in your case you are considering x<0 and y>0 so
x+|x|+y=7 x+|y|-y=6
x+-x+y=7 x+y-y=6
$$y=7$$ $$x=6$$

but $$x=6$$ is not a valid option because you are considering negative values for x.

If x is negative ONLY |x|=-x you cannot change the sign of all Xs
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14 Jun 2013, 07:59
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WholeLottaLove wrote:
Ok, so here is my question:

Let's say we look for all the cases of x,y being positive and negative (four in total)

(x is negative)

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

If x is negative, why don't we plug in -x for all values of x in both equations (i.e. x+|x|+y=7 ==> -x+-x+y=7)

Is 2x + 7 = 0 same as -2x + 7 = 0?

You know that these two are different equations and yield different values of x. x can be negative/positive.

On the other hand, how do you solve something like this: 2|x| + 7 = 0
You need the value of x, not of |x|. How will you get the value of x?

You will use the definition of |x|

|x| = x if x >= 0
|x| = -x if x < 0

So you take 2 cases:

Case 1:
x >= 0 so |x| = x
2|x| + 7 = 0 => 2x + 7 = 0
x = -7/2 (this doesn't work since x must not be negative)

Case 2:
x < 0 so |x| = -x
2|x| + 7 = 0 => 2(-x) + 7 = 0
x = 7/2 (this doesn't work either since x must be negative)

So there is no value of x that satisfies 2|x| + 7 = 0
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Re: If x+|x|+y=7 and x+|y|-y=6 what is x+y=?   [#permalink] 14 Jun 2013, 07:59

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# If x+|x|+y=7 and x+|y|-y=6 what is x+y=?

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