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# What is the distance between x and y on the number line?

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What is the distance between x and y on the number line? [#permalink]

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25 Mar 2012, 17:30
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What is the distance between x and y on the number line?

(1) |x| – |y| = 5
(2) |x| + |y| = 11
[Reveal] Spoiler: OA

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Re: What is the distance between x and y on the number line? [#permalink]

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25 Mar 2012, 23:22
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What is the distance between x and y on the number line?

Question: |x-y|=?

(1) |x| – |y| = 5. Not sufficient: consider x=10, y=5 and x=10, y=-5.
(2) |x| + |y| = 11. Not sufficient: consider x=10, y=1 and x=10, y=-1.

(1)+(2) Solve the system of equation for |x| and |y|: sum two equations to get 2|x|=16 --> |x|=8 --> |y|=3. Still not sufficient to get the single numerical value of |x-y|, for example consider: x=8, y=3 and x=8, y=-3. Not sufficient.

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Re: What is the distance between x and y on the number line? [#permalink]

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26 Mar 2012, 00:30
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Solving the two equations will give x as 8 and y as 3. But since mod sign is there, x and y can take any value, either positive or negative. Hence, both the statements are insufficient.

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Re: What is the distance between x and y on the number line? [#permalink]

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18 Dec 2012, 09:16
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Bunuel can you clarify what can be wrong below approach?

-------------------------------------
By multiplying statements 1 & 2
(1) |x| – |y| = 5
(2) |x| + |y| = 11

$$X^2-Y^2=55$$

i.e. $$(x+y)(x-y)=55 = 11 * 5 = - 11 * - 5$$ (i.e. both factors are either positive or negative)

Hence only two possible solutions for this – i.e. either (x=8 & y=3) OR (x= -8 & y= -3)
In both cases the distance between them is 5.

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Re: What is the distance between x and y on the number line? [#permalink]

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23 Dec 2012, 06:55
PraPon wrote:
Bunuel can you clarify what can be wrong below approach?

-------------------------------------
By multiplying statements 1 & 2
(1) |x| – |y| = 5
(2) |x| + |y| = 11

$$X^2-Y^2=55$$

i.e. (x+y)(x-y)=55 = 11 * 5 = - 11 * - 5 (i.e. both factors are either positive or negative)

Hence only two possible solutions for this – i.e. either (x=8 & y=3) OR (x= -8 & y= -3)
In both cases the distance between them is 5.

(x+y)(x-y)=55 does not mean that either (x=8 & y=3) OR (x= -8 & y= -3). There are more integer solutions (for example x=+/-28 and y=+/-27) and infinitely many non-integer solutions.
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Re: What is the distance between x and y on the number line? [#permalink]

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23 Dec 2012, 13:41
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Re: What is the distance between x and y on the number line? [#permalink]

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16 Jul 2013, 23:22
From 100 hardest questions
Bumping for review and further discussion.
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Re: What is the distance between x and y on the number line? [#permalink]

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18 Jul 2013, 11:56
What is the distance between x and y on the number line?

(1) |x| – |y| = 5

|11|-|6|=5
Distance is five

|11|-|-6|=5
Distance is seventeen
INSUFFICIENT

(2) |x| + |y| = 11

|5|+|6| = 11
Distance is one

|5|+|-6| = 11
Distance is negative eleven

INSUFFICIENT

This problem, to me, seems much easier than a 700 level question. a and b provide us with multiple valid values for x and y, none of which entirely (i.e. are the same) Can someone tell me if I am oversimplifying this problem? Thanks!

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Re: What is the distance between x and y on the number line? [#permalink]

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24 Jun 2015, 09:51
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Re: What is the distance between x and y on the number line? [#permalink]

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31 Mar 2016, 17:37
calreg11 wrote:
What is the distance between x and y on the number line?

(1) |x| – |y| = 5
(2) |x| + |y| = 11

i picked 8 and 3
1. x=8, y=3 -> 5 or x=-8, y=3 -> distance is 11. 1 alone is insufficient.
2. x=8, y=3 -> 5 or x=-8, y=3 -> distance is 11. 2 alone is insufficient.

1+2
same info from 1 and 2. C is out, and the answer must be E.

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Re: What is the distance between x and y on the number line? [#permalink]

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06 Apr 2016, 21:57
Bunuel wrote:
What is the distance between x and y on the number line?

Question: |x-y|=?

(1) |x| – |y| = 5. Not sufficient: consider x=10, y=5 and x=10, y=-5.
(2) |x| + |y| = 11. Not sufficient: consider x=10, y=1 and x=10, y=-1.

(1)+(2) Solve the system of equation for |x| and |y|: sum two equations to get 2|x|=16 --> |x|=8 --> |y|=3. Still not sufficient to get the single numerical value of |x-y|, for example consider: x=8, y=3 and x=8, y=-3. Not sufficient.

Hi... I have a question when we solve both the eqs tog we get two values for |y|=3 and -13. Right?

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Re: What is the distance between x and y on the number line? [#permalink]

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06 Apr 2016, 23:33
enasni wrote:
Bunuel wrote:
What is the distance between x and y on the number line?

Question: |x-y|=?

(1) |x| – |y| = 5. Not sufficient: consider x=10, y=5 and x=10, y=-5.
(2) |x| + |y| = 11. Not sufficient: consider x=10, y=1 and x=10, y=-1.

(1)+(2) Solve the system of equation for |x| and |y|: sum two equations to get 2|x|=16 --> |x|=8 --> |y|=3. Still not sufficient to get the single numerical value of |x-y|, for example consider: x=8, y=3 and x=8, y=-3. Not sufficient.

Hi... I have a question when we solve both the eqs tog we get two values for |y|=3 and -13. Right?

|y| is an absolute value of y, so it cannot be negative. When we solve for |y|, we get that |y| = 3, so y = 3, or y = -3.

Hope it's clear.
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Re: What is the distance between x and y on the number line? [#permalink]

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22 May 2017, 18:24
calreg11 wrote:
What is the distance between x and y on the number line?

(1) |x| – |y| = 5
(2) |x| + |y| = 11

Picking numbers works well here. The trick is to realize that you can simply make either x or y negative to install a new case to test for sufficiency.

GOAL: What is the distance between x and y? Must be one discrete value, not multiple.

Statement 1: Pick 11 and 6. 11-6 = 5, so this case matches the given information and the distance is 5. Now you can make either 11 or 6 negative, so set x = -11. Now abs(-11) - 6 = 5, but the distance is 17 units. So this case is not sufficient because we have multiple possible distances on the number line.

Statement 2: Here abs(x) + abs(y) = 11. Pick 5 and 6, the most obvious choices. The distance between them is 1. Now turn 5 into -5, and the statement is still valid (abs(-5) + abs(6) =11, but their distance is now 11. Not sufficient.

Combined they are not sufficient. You can test via 8 = x, 3 = y and then -8 = x, 3=y.

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Re: What is the distance between x and y on the number line? [#permalink]

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22 May 2017, 18:47
calreg11 wrote:
What is the distance between x and y on the number line?

(1) |x| – |y| = 5
(2) |x| + |y| = 11

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Condition (1)
In the case that $$x = 6$$, $$y = 1$$, the distance $$x$$ and $$y$$, $$| x - y | = 5$$.
In the case that $$x = 6$$, $$y = -1$$, the distance $$x$$ and $$y$$, $$| x - y | = 7$$.
Thus we don't have a unique solution.

Condition (2)
In the case that $$x = 6$$, $$y = 5$$, the distance $$x$$ and $$y$$, $$| x - y | = 1$$.
In the case that $$x = 6$$, $$y = -5$$, the distance $$x$$ and $$y$$, $$| x - y | = 11$$.

Thus we don't have a unique solution.

Condition (1) & (2)
If we add two equation, we have $$2|x| =16$$ or $$|x| = 8$$. Thus $$x = \pm 8$$.
If we subtract the first equation from the second one, we have $$2|y| =6$$ or $$|y| = 3$$. Thus $$y = {\pm}3$$.

In the case that $$x = 8$$ and $$y = 3$$, the distance $$x$$ and $$y$$, $$| x - y | = 5$$.
In the case that $$x = 8$$ and $$y = -3$$, the distance $$x$$ and $$y$$, $$| x - y | = 11$$.
Thus we don't have a unique solution.

Normally for cases where we need 2 more equations, such as original conditions with 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, we have 1 equation each in both 1) and 2). Therefore C has a high chance of being the answer, which is why we attempt to solve the question using 1) and 2) together. Here, there is 70% chance that C is the answer, while E has 25% chance. These two are the key questions. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer according to DS definition, we solve the question assuming C would be our answer hence using 1) and 2) together. (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: What is the distance between x and y on the number line?   [#permalink] 22 May 2017, 18:47
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