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# If |x−y|=y, what is the value of x?

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Math Expert
Joined: 02 Sep 2009
Posts: 53768
If |x−y|=y, what is the value of x?  [#permalink]

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07 Mar 2016, 09:17
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66% (01:38) correct 34% (01:42) wrong based on 306 sessions

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If |x−y|=y, what is the value of x?

(1) xy>0
(2) y=6

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Math Expert
Joined: 02 Aug 2009
Posts: 7424
Re: If |x−y|=y, what is the value of x?  [#permalink]

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07 Mar 2016, 19:32
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Bunuel wrote:
If |x−y|=y, what is the value of x?

(1) xy>0
(2) y=6

Hi,

lets see what info can we get from Q

|x−y|=y
inference:
1)y is positive
2) x= 0 or x=2y

lets see the statement

(1) xy>0
this tells us that xis NOT equal to 0..
we already know both x and y are positive
so x=2y..
we do not know the NUMERIC value of y..
Insuff

(2) y=6
this tells us either x=2y=12 or x=0..
two values
Insuff

Combined
X is not 0, so x=12
Suff

C

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Re: If |x−y|=y, what is the value of x?  [#permalink]

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22 Sep 2017, 01:01
Bunuel wrote:
If |x−y|=y, what is the value of x?

(1) xy>0
(2) y=6

|x−y| means the LHS is Positive.....Thus, y is positive

1) xy>0

Let x =2, y = 1

Let x = 4, y = 2

Insufficient

2) y=6

|x−6|=6

x = 12 or x =-6

Insufficient

Combine 1 & 2

y is positive and according to Statement 1, x must be positive

We get one solution x =12

Manager
Joined: 24 Sep 2018
Posts: 140
Re: If |x−y|=y, what is the value of x?  [#permalink]

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25 Sep 2018, 12:16
Using the two-case rule for absolute values, you can restate the information in the stimulus as:

Quote:
Either x – y = y
Or x – y = -y

Then performing the algebra in each case, the first tells you that x = 2y, and the second tells you that x = 0.
Those are then the two possibilities for x: either x = 0, or it equals 2y.

Quote:
Statement 1 tells you that x cannot be 0, so you know that x = 2y. But since you don’t know y, you cannot determine x.

Quote:
Statement 2 tells you that y = 6, which means that x = 12…or x could still be 0. Taken together, the statements guarantee that x = 12, as you know from statement 2 that it’s either 12 or 0, and from statement 1 that it’s not 0.

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Re: If |x−y|=y, what is the value of x?  [#permalink]

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25 Sep 2018, 13:07
Bunuel wrote:
If |x−y|=y, what is the value of x?

(1) xy>0
(2) y=6

VERY beautiful problem, Bunuel. (Kudos!)

$$? = x$$

$$\left| {x - y} \right| = y\,\,\,\,\,\, \Rightarrow \,\,\,\,\,y \geqslant 0\,\,\,\,\,\,\,\,{\text{AND}}\,\,\,\,\,\,\,\left\{ \begin{gathered} \,\,x = 0\,\,\,,\,\,\,y \geqslant 0\,\,\,{\text{free}} \hfill \\ \,\,{\text{OR}}\,\,\,\, \hfill \\ 0 \ne x\,\,,\,\,\,{\text{0}}\,\,\mathop {\text{ < }}\limits^{\left( * \right)} \,\,{\text{dist}}\left( {x,y} \right) = {\text{dist}}\left( {y,0} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,0 < y < x = 2y \hfill \\ \end{gathered} \right.$$

$$\left( * \right)\,\,\,0 \ne x = y\,\,\,\,\, \Rightarrow \,\,\,\,\,0 = \left| {x - y} \right| = y = x\,\,\,\,{\text{impossible}}\,\,\,$$

$$\left( 1 \right)\,\,\,xy > 0\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {x,y} \right) = \left( {2,1} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{2}}\,\, \hfill \\ \,{\text{Take}}\,\,\left( {x,y} \right) = \left( {4,2} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{4}}\,\, \hfill \\ \end{gathered} \right.$$

$$\left( 2 \right)\,\,\,y = 6\,\,\,\,\left\{ \begin{gathered} \,{\text{Take}}\,\,\left( {x,y} \right) = \left( {0,6} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{0}}\,\, \hfill \\ \,{\text{Take}}\,\,\left( {x,y} \right) = \left( {12,6} \right)\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,{\text{ = }}\,\,{\text{12}}\,\, \hfill \\ \end{gathered} \right.$$

$$\left( {1 + 2} \right)\,\,\,\,\,\left\{ \begin{gathered} \,x \ne 0 \hfill \\ \,y = 6 \hfill \\ \end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = x = 2y = 12\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Re: If |x−y|=y, what is the value of x?   [#permalink] 25 Sep 2018, 13:07
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