Problem Solving

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

AnuragRatna 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

Since x|x| < x
x|x|-x < 0
i.e. x(|x|-1) < 0
i.e. either

x > 0 and |x|-1 < 0
i.e. x > 0 and |x| < 1
i.e. either 0 < x < 1

OR
x < 0 and |x|-1 > 0
i.e. x < 0 and |x| > 1
i.e. x < -1

No Option Matches... Incorrect Question

AnuragRatna 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

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.

AnuragRatna wrote: The question corrected by changing x|x|<x into x / |x|<x

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

x /|x| < x
i.e. x < |x| x
i.e. |x|*x - x > 0
i.e. x(|x|-1) > 0

i.e. Either

x>0 as well as |x|-1 > 0
x>0 as well as |x|>1
i.e. x>1

OR

x<0 as well as |x|-1 < 0
x<0 as well as |x|<1
i.e. 0 < x < -1

Answer: Option B