Data Sufficiency on Absolute value

[Joepc]

The Question:-
Is |x-6| >1

Statement 1 |x-4| >2
Statement 2 |x-7|>3

As per the Question stem
The Distance from the point 6 to X to be greater than 1 ,
so when |x-6| is positive (right direction in number line),X can take any value greater than 7(x>7)
and When |x-6| is Negative(left direction in number line),X can take any value less than 5(x<5)

Taking First Statement |x-4| >2
so when |x-4| is positive ,X can take any value greater than 6(x >6) This will not give definite answer for the question because it is greater than 6 and you can have values from 6.1,6.2.. and the distances will be 0.1,0.2.. it is not greater than 1 always from the right direction of 6 in number line

and and When |x-4| is Negative,X can take any value less than 2(x<2)This will give a definite answer to the question as any value less than 2 will make the distance greater than 1 from the left direction of 6 in number line

Since the right side is not giving valid answer , I consider Statement 1 as Insufficient

In a Similar way examining the Statement 2, the Range falls X>10 and X<4 and any value greater than 10 and less than 4 will help me to give a definite answer to the question which will be always greater than 1.
in both left side and right side of 6

So i consider the statement 2 as sufficient

Am i correct?, If i am wrong please guide me.

[GMATGuruNY]

Is |x-6| >1

Statement 1 |x-4| >2
Statement 2 |x-7|>3

|a-b| = the DISTANCE BETWEEN a and b.

Question stem, rephrased:
Is the distance between x and 6 greater than 1?
On the number line:
<---x---5...6...7---x--->
Is x within one of the blue ranges above?

Statement 1:
The distance between x and 4 is greater than 2, as follows:
<---x----2...4...6---x--->
If x=1, then x is within one of the blue ranges above.
If x=6.1, then is NOT within one of the blue ranges above.
INSUFFICIENT.

Statement 2:
The distance between x and 7 is greater than 3, as follows:
<---x---4...7...10---x--->
Whether x is to the left of 4 or to the right of 10, it will be within one of the blue ranges above.
SUFFICIENT.

The correct answer is B.