Question stem: x not equal to 0, |x| < 1?, or -1 < x < 1?
1) x/|x| < x, since we know |x| is positive (and not equal to 0), we can multiply:
x < x|x|
x - x|x| < 0
x(1-|x|) < 0
When we have the form xy < 0, we know that either x is positive and y is negative, or x is negative and y is positive.
Case 1: x < 0 and 1-|x| > 0, or |x| < 1, or -1 < x < 1. Combined this says that -1 < x < 0.
Case 2: x > 0 and 1-|x| < 0, or |x| > 1. Combined this means x > 1.
Taking case 1 OR case 2, -1 < x < 0 OR x > 1. The question stem requires -1 < x < 1, so this statement is insufficient.
2) |x| > x, or x < 0. x could be -1,000,000 or -1/2. Insufficient.
1 & 2) Combining the second statement with the first, we can take case 1, which stated that -1 < x < 0, sufficient.
C
That was the classical solution, but I want to present an alternative that can save a LOT of time on this type of question:
Sketch the a graphs of each expression in the statements as a function.
1) y = x/|x|, this is a straight horizontal line with y = -1 for x < 0, and y = 1 for x > 0.
Draw in the graph for y = x. You can see easily that x/|x| is less than x for -1 < x < 0 and for all x > 1, so the statement is not sufficient.
2) Draw y = |x| and y = x. If you can't pull "x is less than 0" from statement 2, this will make it clear very quickly. From the drawing you can see that any negative number satisfies this inequality, so the statement is not sufficient.
