Question rephrased: Is x between -1 and 1?If x is not equal to 0, is |x| less than 1?
(1) x/|x|< x
(2) |x| > x
Statement 1: 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.
Case 1: x<-1
Plug x = -2 into x/|x| < x:
-2/ |-2| < -2
-1 < -2.
Doesn't work.
Thus, x < -1 is not a valid range.
Case 2: -1<x<0
Plug x = -1/2 into x/|x| < x:
-1/2/ |-1/2| < -1/2
-1 < -1/2.
This works.
Thus, -1<x<0 is a valid range.
Case 3: 0<x<1
Plug x = 1/2 into x/|x| < x:
(1/2)/ |1/2| < 1/2
1 < 1/2
Doesn't work.
Thus, 0<x<1 is not a valid range.
Case 4: x>1
Plug x = 2 into x/|x| < x:
2/ |2| < 2
1 < 2.
This works.
Thus, x > 1 is a valid range.
Thus, -1<x<0 or x>1.
INSUFFICIENT.
Statement 2: |x| > x
Any negative value will satisfy this inequality.
If x=-1/2, then x is between -1 and 1.
If x=-2, then x is NOT between -1 and 1.
INSUFFICIENT.
Statements combined:
The only range that satisfies both statements is -1<x<0.
Thus, x is between -1 and 1.
SUFFICIENT.
The correct answer is C.
