If X/|X| < X Which of the following must be true about X ?
a) X>1
b) X>-1
c) |X| < 1
d) |X| = 1
e) |X|^2 > 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.
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.
Since it's possible that 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.
Eliminate E, since it doesn't have to be true that |x|² > 1.
Since it's possible that x=2:
Eliminate C, 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.
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