Question

Let, (x) = Least integer greater than or equal to x
[x] = Greatest integer less than or equal to x
|x| = absolute value of x,

Which of the following always holds good if x < 0?

• A. [|x|] = | [x ] |
• B. [ | x | ] < | [ x ] |
• C. [ | x | ] > | [ x]|
• D. | x |< [ x ]

Answer 1

The correct option is B

[ | x | ] < | [ x ] |

The right answer is $\left[|x|\right]<|\left[x\right]|$.
Let x = n + f where n is its integral part & f is its fractional part.
Hence, | x | = n + f if x is positive and | x | = - n - f if x is negative.
If x is - ve, then | x | = -n -f .
Hence,[ | x | ] = -n and | [x] | is always non negative.
Hence the alternative (2) holds good.

Method 2- Using assumption
Take a value say x= -1.4 (you can take anything!)
Thus, | - 1.4| = 1.4 and [1.4] = 1
Also, [ -1.4] = -2 and |-2| = 2

Glance at the answer options; the only one satisfying this assumption is $\left[|x|\right]<|\left[x\right]|$.