For ｜x｜+｜x-1｜=1 to be true the x has to be equal to 0 or positive but less than 1.

Statement 1 says x > = 0 . Considering 0 , the statement is true . But consider x = 2 .

|2| + |2-1| is not equal to 1 . hence insufficient .

Statement 2 says x < = 1 . Again not sufficient . You may consider values such as 1 or - 0.5 for authentication purpose .

Taking both together , 0 < = x < = 1 . This satisfies the condition that I cited initially .

Hence OA C

The prompt can be rephrased as

If x is positive, we have Is x=1 or If x is negative , we have Is x=0. Is x=1 or is x=0

Statement 1 tells us that x=0 or 1 or greater than 1. Insufficient
Statement 2 tells us that x=1 or 0 or less than 1. Insufficient
Both one and two give us 0<=x<=1. Sufficient

One more thing, I suspect the prompt may have also mentioned that x is in integer, otherwise the equation will fail for positive fractional values between 0 and 1

One more thing, I suspect the prompt may have also mentioned that x is in integer, otherwise the equation will fail for positive fractional values between 0 and 1

That's not the case:

x = 1/4
｜x｜ + ｜x-1｜= ｜1/4｜ + ｜-3/4｜ = 1/4+ 3/4 = 1

In general, if 0 <= x <= 1, then |x-1| = 1-x, so
｜x｜ + ｜x-1｜ = x + (1-x) = 1

IMO : C

in fact each statement is a condition of each part of expression in the question that alone can not be sufficient. stat 1 tell us that the first expression come out of absolute value in positive form and second statement tell us that the second expression come out in negative mode (bcuz each one of them is critical points of each stats) ==> x-x+1=1 ==> together sufficient