Since |x| can never be negative, it's safe to multiply both sides of the inequality by |x|.
This gives us
x < x * |x|
Now we have two cases.
Case I:: x > 0
In this case, we divide both sides by x, and we get
1 < |x|
Case II:: x < 0
In this case, we divide both sides by x again, but this time the sign FLIPS, since we're dividing by a negative.
1 > |x|
Remember that x is a negative number. If x is negative, |x| = -x. So we REALLY have
1 > -x
or
-1 < x
Since x is negative, we can write this as -1 < x < 0, and we're done!
So our solution sets are
x > 1
and
0 > x > -1
The only answer that describes both sets is B. (Note that we can't have 1 ≥ x ≥ 0, but B doesn't say that we do; it only says that x, whatever it is, must be greater than -1 - it doesn't say that x can take any value greater than -1.)
