Tricky absloute question

Mo2men wrote:If x≠0 and x/|x|<x, which of the following must be true?

A. x>1
B. x>−1
C. |x|<1
D. |x|>1
E. −1<x<0

Prove that four of the five answer choices do NOT have to be true.
Constraint: x/|x| < x

It's possible that x=2, since 2/|2| < 2, with the result that C and E do not have to be true.
Eliminate C and E.
It's possible that x=-1/2, since (-1/2)/|-1/2| < -1/2, with the result that A and D do not have to be true.
Eliminate A and D.

The correct answer is B.

Algebra:

x/|x| < x

x < x|x|

0 < x|x| - x

0 < x (|x| - 1)

The CRITICAL POINTS are -1, 0 and 1.
These are the only values where x(|x|-1) = 0.
To determine the ranges where x(|x|-1) > 0, test one value to the left and right of each critical point.

Plug x = -2 into x/|x| < x:
-2/ |-2| < -2
-1 < -2.
Doesn't work.
x < -1 is not a valid range.

Plug x = -1/2 into x/|x| < x:
-1/2/ |-1/2| < -1/2
-1 < -1/2.
This works.
-1<x<0 is a valid range.

Plug x = 1/2 into x/|x| < x:
(1/2)/ |1/2| < 1/2
1 < 1/2
Doesn't work.
0<x<1 is not a valid range.

Plug x = 2 into x/|x| < x:
2/ |2| < 2
1 < 2.
This works
x > 1 is a valid range.

Thus, the valid ranges are -1<x<0 and x>1.

Case 1: x=2
Eliminate C, since it doesn't have to be true that |x|<1.
Eliminate E, since it doesn't have to be true that -1<x<0.

Case 2: x=-1/2
Eliminate A, since it doesn't have to be true that x>1.
Eliminate D, since it doesn't have to be true that |x|>1.

The correct answer is B.

Since both -1<x<0 and x>1 are to the right of -1, it must be true that x > -1.