If x ≠0, then what is the value of (|x|)/x ?

From the question:
if \(x < 0\) then \(\frac{(|x|)}{x} = -1\) since you would have a \(\frac{pos}{neg}\)

if \(x > 0\) then \(\frac{(|x|)}{x} = 1\) since you would have a \(\frac{pos}{pos}\)

So this question boils down to is x pos or neg?

Statement 1) \(\sqrt{(x^2)}=x\) This means that x must be a positive value so this answers the question. Sufficient

The square root function returns a positive value. If you find yourself screaming or wondering "What about the negative value?!?" Look here https://gmatclub.com/forum/math-number-theory-88376.html#p666609 and do a search on the page for "even roots" (without quotes).

Statement 2) \(|x-4| = \frac{x}{3}\)
Pulling off the absolute value signs gives us two possibilities:
A) \(x - 4 = \frac{-x}{3}\) and B) \(x-4 = \frac{x}{3}\)
Take a look at each one:
A) \(x-4 = \frac{-x}{3}\)
\(3x-12 = -x\)
\(4x = 12\)
\(x = 3\) so x is positive and that answers the question.

B)\(x - 4 = \frac{x}{3}\)
\(3x - 12 = x\)
\(2x = 12\)
\(x = 6\) so x is positive and that answers the question.

Both A & B give me the same answer to the question (positive one) so 2 is sufficient.

Statement 1 and 2 are both sufficient so the answer is D.