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# On the number line, are the points x and y on the same side

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Intern
Joined: 15 Apr 2014
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On the number line, are the points x and y on the same side  [#permalink]

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06 Jul 2014, 23:50
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50% (01:29) correct 50% (01:36) wrong based on 182 sessions

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On the number line, are the points x and y on the same side of zero?

(1) x and y are equidistant from zero
(2) The sum of the distances from x to 1 and from y to 1 is less then 1.

according to what I found online the answer is B but I dont get it...
B. From statement 1, x can be equal to y or x=-y  insufficient. From
statement 2, x and y are less than 2 and more than 0  x and y are both
positive, sufficient.

I think it should be E
statement 2: if x=2 and y=-0.5 the the distance from x to 1 is 1 and the distance from y to 1 is -1.5
the distances sum is 1-1.5=-0.5 which is less then1

can someone plz explane ?
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Re: On the number line, are the points x and y on the same side  [#permalink]

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07 Jul 2014, 02:54
1
The distance always has to be positive. if one of the points would be negative the distance of that point to 1 would be greater than one. That means that the sum of the two distances (remember that distances are always positive) would be greater than 1, too. Therefore, the two points must lie between 0 and one and both have to be positive.

E.g. let's say that x = -0.0001. its distance to 1 would be 1.0001 so the sum of the distances of x and y would be greater than 1.

I hope it helps.
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Re: On the number line, are the points x and y on the same side  [#permalink]

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09 Sep 2014, 00:29
Is it wrong to assume that x and y are two different points? If not, then why is D incorrect? x and y are equidistant means both will not be on the same side.
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Re: On the number line, are the points x and y on the same side  [#permalink]

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09 Sep 2014, 08:03
kamranjkhan wrote:
Is it wrong to assume that x and y are two different points? If not, then why is D incorrect? x and y are equidistant means both will not be on the same side.

Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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Re: On the number line, are the points x and y on the same side  [#permalink]

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19 Nov 2015, 20:46
Draw the actual number line:

We have |x-1| + |y-1| < 1, so the number line can look like:

0------y---x--1--x---y------2

Even if x = y we would still have 0<x,y<2. So x,y at the same side of the number line.
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Re: On the number line, are the points x and y on the same side  [#permalink]

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20 Nov 2015, 05:39
Hi Bunuel ,

Please post a detail solution and Concept for the same .
Thanks .
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Regards ,

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Re: On the number line, are the points x and y on the same side  [#permalink]

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21 Nov 2015, 03:09
abhisheknandy08 wrote:
Hi Bunuel ,

Please post a detail solution and Concept for the same .
Thanks .

On the number line, are the points x and y on the same side of zero?

(1) x and y are equidistant from zero. Both x and y can be 1 or x can be 1 and y can be -1. Not sufficient.

(2) The sum of the distances from x to 1 and from y to 1 is less then 1. This translates to |$$x - 1| + |y - 1| < 1$$. So, we have that the sum of two absolute values, the sum of two non-negative values is less than 1. Now, can x be negative or 0? No, because if it's negative or 0, then $$|x - 1| \geq{1}$$ and in this case $$|x - 1| + |y - 1| =(something \ more \ than \ or \ equal \ to \ 1) + (something \ more \ than \ 0) > 1$$, which contradicts given statement. The same way y cannot be negative or 0. Therefore both x and y must be positive. Sufficient.

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Re: On the number line, are the points x and y on the same side  [#permalink]

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22 Nov 2015, 02:26
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.

On the number line, are the points x and y on the same side of zero?

(1) x and y are equidistant from zero
(2) The sum of the distances from x to 1 and from y to 1 is less then 1.

2) -------------------|-------------|---------|-----|------------------yes
0 1/2 x(=y) 1
-------------------|-------------|-------------|-|-|------------- yes
0 1/2 x 1 y

There are 2 variables (x,y) and 2 equations are given by the conditions, so there is high chance (C) will be the answer.
Looking at the conditions together,
x=y and both are numbers close to 1.
For example, x=y=0.7, and the answer seems like (C), but this is a commonly made mistake.
Looking at them separately,
For condition 1, the answer is 'yes' for x=y=1, but 'no' for x=-1, y=1. So this is insufficient.
For condition 2, x=y=0.9 or x=0.9 and y=1.1. This is always 'yes' and is sufficient.

the original question is

On the number line, are 0 between the points x and y?

(1) The distnace between x and 0 is equal to the distance between y and 1
(2) The sum of the distances from x to 0 and from y to 1 is less then 1.

There are 2 variables (x,y) and 2 equations are given by the conditions, so there is high chance (C) will be our answer.
Looking at them together, the answer is 'yes' for (x,y)=(1/4,3/4),
but 'no' for (x,y)=(-1/4,3/4) .
This is insufficient, so the answer becomes (E).

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, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. 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 using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: On the number line, are the points x and y on the same side  [#permalink]

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07 Jul 2018, 04:09
Hi Bunuel,

I understand this equation - |x−1|+|y−1|<1. Logically it makes sense to say that both x and y are positive, but I'm unable to solve this mathematically. Acc to me, the equation is as follows

-1<x-1+y-1<1 so -1+2<x+y<1+2 1<x+y<3. This just proves that the sum of x+y is positive, but considering this alone how can we be sure that individual values of x and y lie on the same side of the number line. This is how I solved and I got it wrong.
Kindly elaborate.
Thankyou
Math Expert
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Posts: 49496
On the number line, are the points x and y on the same side  [#permalink]

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07 Jul 2018, 08:28
Hi Bunuel,

I understand this equation - |x−1|+|y−1|<1. Logically it makes sense to say that both x and y are positive, but I'm unable to solve this mathematically. Acc to me, the equation is as follows

-1<x-1+y-1<1 so -1+2<x+y<1+2 1<x+y<3. This just proves that the sum of x+y is positive, but considering this alone how can we be sure that individual values of x and y lie on the same side of the number line. This is how I solved and I got it wrong.
Kindly elaborate.
Thankyou

If $$x \leq 0$$, then x - 1 will be negative. We know that if $$a \leq 0$$, then $$|a| = -a$$. So, if $$x - 1 < 0$$, then $$|x - 1| = -(x - 1)$$. So, we'd get $$-(x - 1) + |y − 1| < 1$$ --> $$-x + |y − 1| < 0$$. Here $$-x = -negative = positive$$ and $$|y − 1| \geq 0$$ (because an absolute value is always positive or 0). But $$(positive) + (non-negative) = positive$$. Thus $$-x + |y − 1| < 0$$ cannot be true and thus $$x \leq 0$$ cannot be true.

The same way we can prove that y <= 0 is not possible.

Therefore, both x and y must be positive.
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On the number line, are the points x and y on the same side &nbs [#permalink] 07 Jul 2018, 08:28
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