If x is a non-zero number, Is x|x|<x? (1) −3x<0 (2) x|x|>0

If x is a non-zero number, Is \(\frac{x}{|x|}<x?\)

(1)\(\frac{−3}{x}<0\). This implies that x is positive. In this case, |x| = x. So, the question becomes: is \(\frac{x}{x}<x?\)? Or, which is the same: is 1 < x. We just know that x is positive: it can be more or less than 1. Not sufficient.

(2) \(\frac{x}{|x|}>0\). This also implies that x is positive (If x were negative, then we'd get x/(-x) = -1, which is less than 0, not more, as given here). x > 0, implies that |x| = x. So, the question, as above, becomes: is \(\frac{x}{x}<x?\)? Or, which is the same: is 1 < x. We just know that x is positive: it can be more or less than 1. Not sufficient.

(1)+(2) Both statements give the same info: x > 0. If 0 < x <= 1, then the answer to the question will be NO but if x > 1, the answer to the question will be YES. Not sufficient.


Answer: E.