This is a good question. Absolute values is a problem area for a lot of students.
Lets look at (1). It tells us that x/ |x| < x. Since |x| is positive we can multiply both sides by it and get x < |x|x. This does not tell us much because if x < 0 and we divide both sides by x we get |x| < 1 (we reverse the direction of the < when we divide by a negative number) However, if x >0 and we divide both sides by it we have |x| > 1. (1) is inconclusive. Now, lets look at (2). Suppose x >0, then x = |x|. However, if x < 0, then x < |x|. In other words, x must be a negative number. Again, the result is inconclusive.
Combining the two statements: Because x < 0 (according to (2)), then |x| < 1 (according to (1)).
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