x/|x|<x, which of the following must be true about x ?

You can rearrange this to:

x < x|x|

If x is positive, then 1 < |x|

If x is negative, the 1 > |x|


Only B fits both of the criteria for both...

While the mathematical explanation provided previously is always good to understand, remember that all the GMAT cares about is whether you can get to the right answer. When it comes to most absolute value questions, one should always try to plug-in the answer choices and/or pick numbers. The key is to know which numbers to pick in this case: -1, -1/2, 0, 1/2, 1, & 2; the same numbers; extreme numbers. In this case, one could pick x = -1/2 for each answer (except for B) to show that they do not have to be true. While this doesn't prove mathematically that B is correct, it does allow the other four answers to be eliminated. This is not a trick - in fact, its what the GMAT expects.

I hope this helps.

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

Doesn't option B fails at x=1/2

manihar.sidharth wrote:Doesn't option B fails at x=1/2

1/2 does not satisfy the condition given in the stem ( x/|x| < x), so it is not a valid possibility.

I found it easier to switch the original equation around a bit. I multiplied both sides by |x|, so you get: x<x|x|. You can do this, because the absolute value of x is always positive. Then, you can look at it in two ways: if x is positive and if x is negative.

If x is positive, then both sides of the inequality will be positive and x has to be greater than 1, because any positive fraction (less than one) multiplied by itself yields a number of smaller value.

If x is negative, then both sides of the inequality will be negative and x must be greater than negative one, because in order for |x|x to be greater than x, it must have a smaller absolute value.

x>1 is just a sub set of x>-1, so your answer for what must be true is x>-1

See Img