|x-a| < |x-b|. Is a<b?
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Updated on: 16 Apr 2017, 19:35
Question Stem: |x−a|<|x−b| Is a<b?
The question stem tells us that the distance of a from x is less than the distance of b from x.
Below are certain possibilities of above.
Case 1
a - - - - - X - - - - - - - -b [Here we see a & b are on either sides of X]
Above X can be on either sides of 0
a - - - -0 - X - - - - - - - -b [Here X is towards the right of 0, making x and b positive, whereas x as negative]
a - - - - - X - 0- - - - - - -b [Here X is towards the left of 0, making x and a negative, whereas b as positive]
a - - - - - X - - - - - - - -b------0 [Here 0 is towards the right of a,x & b, making x, b and a negative]
Case 2
X - - - - - a - - - - - - - -b [Here we see a & b are on same side of X]
Above X can be on either sides of 0
0 - - - -a - X - - - - - - - -b [Here 0 is towards the left of a,x & b, making x, a and b positive]
a - X - - - - - -b--0 [Here 0 is towards the left of a,x & b, making x, a and b negative]
X - -0- - - a - - - - - - - -b [Here 0 is towards the right of x but left of a & b, making x negative and a&b positive]
So we see how a, b & x can take different positive & negative values.
Question now asks whether a < b?
So as per the above scenarios, we need to know where a is located relative to X and B.
Statement 1 : ab<0
This statement tells us that either a or b is less than 0 or is towards the left of 0
Case 1
b-------0----a---x
Here clearly a is greater than b
Case 2
a---0----x----------b
Here clearly b is greater than a
As we have two different scenarios here, this statement is not sufficient.
Statement 2: For all x>0, |x−a|=|x|+|a|
This statement tells us that X is greater than 0 or towards the right of 0.
Also it tells that distance of a from x is equal to the sum of the distances of X from 0 and a from 0.
Let us assume A is positive, we show this as
0-----a--x--------b
In this case the distance of a from x is 2 (2 dash).
Distance of a from 0 is 5 (5 dash)
Distance of X from 0 is 7 (7 dash)
So given statement says
|x−a|=|x|+|a|
|x−a| = Distance of a from x = 2
|x|= Distance of X from 0 = 7
|a|= Distance of a from 0 = 5
Clearly the equation does not hold up, which tells us that, for the eqaution to hold up, x has to be on the left of 0, ie. x is negative.
Now that we know a is negative and X is positive. Let's see if a < b
Case 1
0-----a--x--------b
Clearly b is greater than a
Case 2
0-----b--a--------x
Clearly a is greater than b
As we have two possibilities here, this statement is not sufficient.
Combining both statements, he is what we get to know
1 - X is positive
2 - a is negative
3 - ab is less than 0, and since a is negative, B has to be positive. Which tells us that a is indeed less than b.
Hence answer is C
Originally posted by
quantumliner on 13 Apr 2017, 12:18.
Last edited by
quantumliner on 16 Apr 2017, 19:35, edited 1 time in total.