Inequality

Check my posts below for two different approaches:

https://www.beatthegmat.com/modulus-ps-t112185.html
https://www.beatthegmat.com/let-get-abso ... 27383.html

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.)