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

[Guest]

I think the answer to this question should be A only
After squaring both sides you get;
x^2 > 1
So 1<x<-1 and the answer to the question stem is always NO..

Can Manhattan Instructors pls confirm if this is correct....
Thanks

[StaceyKoprince]

if x is not equal to zero, is lxl < 1?

Statement 1: [x/lxl] < x
You can't just square both sides because there are two possibilities here: that x/lxl = 1 or that x/lxl = -1. You've just lost one of the possibilities (-1) by squaring both sides. That's why you're only getting one answer, the one that equates to the positive case. See below for the correct manipulations.

You can multiply both sides by lxl to get x < xlxl (Note: don't have to switch signs b/c absolute value of anything is never negative.)

Then, you can divide the plain x on the right side BUT you have to remember there are two possibilities: when x is positive and when x is negative.
if x is positive, then x/x < lxl or 1 < lxl, in which case we answer the question no.
if x is negative, then x/x > lxl or 1 > lxl, in which case we answer the question yes.

These answers contradict, so the statement is insufficient.