statement (1)
for EQUATIONS involving absolute value, like this one, the key realization is that the absolute value of a quantity can signify either the quantity itself or the opposite of the quantity. therefore, if you try each of the sign combinations (pos/neg) of the absolute values in the problem, you'll be guaranteed to find all of the solutions.
(note: in what follows, "+" means leaving the expression within the absolute value bars alone; "-" means reversing the sign of that expression)
in this equation, there are ostensibly four sign combinations, +/+, +/-, -/+, -/-, but it's only necessary to try two of them:
** first, either +/+ or -/-, in which both or neither of the absolute value expressions are flipped. may as well go with +/+ (i.e., leaving both of the absolute value expressions alone while removing the bars): x + 1 = 2(x - 1), or x = 3. plugging this back into the original equation shows that it works.
** second, either +/- or -/+, in which one of the absolute value expressions is flipped. let's go (at random) with flipping the first one: -x - 1 = 2(x - 1), or x = 1/3. plugging this into the original equation also shows that it works.

therefore, statement 1 means that x = 3 or x = 1/3.

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statement (2)

two ways to interpret absolute value inequalities like this one:
** memorize the template of the solution (preferred for efficiency's sake): you should just know that |expression| > a means "either expression > a or expression < -a".
** conceptualize absolute value as distance: in this case, |x - 3| means the distance between x and 3. therefore, this statement means that the distance between x and 3 is greater than 0 (in either direction).
either of these interpretations means that x < 3 or x > 3, or, equivalently, x is not equal to 3.

statement 1 is insufficient, because 1/3 gives a "yes" and 3 gives a "no". statement 2 is also insufficient, because every number except 3 is possible. taken together, though, the two statements are sufficient because they yield a unique value, 1/3, for x.
notice that there's no reason even to figure out whether 1/3 gives "yes" or "no" at this point; it's one value, meaning that it is guaranteed to be sufficient no matter what the answer.

answer = c