Inequality Tough question

rakeshd347 wrote:If x is not equal to 0, is |x| less than 1?

(1) x/|x|< x

(2) |x| > x

Simplify:
x ≠ 0

Is |x| < 1
or
Is -1 < x < 1 ? ... (i)

Solve:

St1:

Verify using very simple numbers:
x = -0.5
-0.5 / 0.5 < -0.5
-1 < -0.5 (i) is true

x = 2
2 / 2 < 2
1 < 2 (i) is false
INSUFFICIENT.

St2:
|x| > x
in other words x = -ive (This means x is always less than |+ive|, one can verify that only -ive values of x will satisfy this equation)
INSUFFICIENT.

St1 + St2:
We know that x is -ive we just need to check other smaller negative values by plugging numbers in the St1. This would narrow down the range if possible.

x = -1
-1 / 1 < -1
-1 < -1 (Invalid)

x = -0.5 (i) is always true as proved in St1.

x = -2
-2 / 2 < -2
-1 < -2 (Invalid)

Thus -1 < x < 0
We can definitely answer (i)

[spoiler]Answer : C[/spoiler]

Regards,
Vivek

To find if |X| < 1

Statement 1:
x/|X| < x

if "x" is positive .. then |X| > 1

if "x" is negative .. then |X| < 1
INSUFFICIENT

Statement 2:
|x| > x
This implies x is surely negative
INSUFFICIENT

Combining..

|X| < 1
SUFFICIENT

Answer [spoiler]{C}[/spoiler]

If x is not equal to 0, is |x| less than 1?

(1) x/|x|< x

(2) |x| > x

Question rephrased: Is x between -1 and 1?

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.