Is |x|< 1?
(1) |x + 1| = 2|x - 1|
(2) |x - 3| ≠ 0
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Given |x|< 1
i.e x<1 and x>-1 i.e basically the question
"Is whether or not -1<x<1"

Now using the choices given --
(1) |x + 1| = 2|x - 1|

Case I :: x+1=2x-2 ---> x=3
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Case II :: -(x + 1)=2(x - 1) ---> -x-1=2x-2
------- i.e 3x=1 --> x=1/3

Case III :: (x + 1)=2(x - 1)
-------- Same as case I ...which implies x=3

Case IV :: (x + 1)=-2(x - 1)
------- x+1=-2x+2 ---> 3x=1 therefore x=1/3

Using the values of x found above i.e 3(no it doesnt lie in the range -1<x<1) and 1/3(yes it lies in the range asked in the question -1<x<1) the answer to the question can either be a yes or no..we are not sure !!

Thus Not Sufficient.

(2) |x - 3| ≠ 0
Using this --> x-3 ≠ 0
i.e x ≠ 3
or -(x - 3) ≠ 0
-x+3 ≠ 0
which implies x ≠ 3
this gives us a infinte value of x which is possible.

Thus not sufficient.

Using 1 and 2
We can defintely answer the question coz we are limited to x = 1/3

Thus Both together is sufficent. !!!



I hope i made sense...!! any comment anyone ?