There's no difference between absolute of x and absolute of absolute of absolute of ... of x.
Allow me to use "[" syntax as an absolute syntax.
x/[[x]]<x
I'll divide this into 2 cases. When x>0 and when x<0
1. When x>0
x/[[x]]<x means
1/[[x]]<1 or [[x]]>1 (divide two sides to a positive number, the inequality sign will not change)
or [x]>1
Thus x>1 or x<-1 However, when this is combined with x>0, the final range of x must be x>1.
2. When x<0
x/[[x]]<x means
1/[[x]]>1 (divide two sides to a negative number, the inequality sign will change) or [[x]]<1
or [x]<1
Thus -1<x<1, however when this is combine with initial condition "x<0", the final range of x: -1<x<0
The final step, combine two range of x of 2 cases:
x>1 and -1<x<0, the result will be x>-1, but excluding the range 0=<x=<1 (Draw all those ranges in value axis for a clearer view, if needed). Now I will try to prove all value in this excluded range acceptable, if I can, the final range will be x>-1
Note at this range 0=<x=<1, there're two integers: 0 and 1.
0 is eliminate as X is differ from 0 ( stated in the hypothesis)
1 remains, and 1 makes the inequation incorrect : 1/[[1]] does not smaller than 1
So the range x>-1 has a value "1" unacceptable. So this range can not be the answer. Let's look at the choices.
The ultimate range must be x>1, however in the answer choice, there's only x>2
The value x = 2 make the inequation correct ( 2/[[2]]<2) so the range x>2 misses a correct value x =2
However, the range x>-1 includes a wrong value x=1
Both are not absolutely correct, but they seem to be the two best possible answers.
Please check the source please.
"There is nothing either good or bad - but thinking makes it so" - Shakespeare.
