The question stem X|X|<2^x can be rewritten as "Is X negative?"
Because, mod X is always positive; hence, the value of the question stem depends entirely on the negative or positive value of x.
Well, that's all we need to consider given the statements in this question. Note that there ARE positive values of x such that x|x| < 2^x ...specifically, if x > 4, this is true.
But yes, if x < 0, the answer to the question is always YES.
2 Explanations