Find the values of x such that (ln|x|/|x|)>(ln|x|/x)

Expert Answers
nathanshields eNotes educator| Certified Educator

`(ln|x|)/|x| > (ln|x|)/x`

First, let's think about the graph of ln|x| (my graph plugin isn't behaving, so graph it yourself if you need to).  The function is positive when x < -1 and also when x > 1, but negative when x is between -1 and 1 (not including -1, 0, and 1).

So let's think of x < -1.  Then the inequality is true, since the left side is positive and the right side is negative.

When x = -1, both sides have 0 on the numerator, so they're equal (inequality is false).

How about x between -1 and 0?  Left side is negative, right side is positive, so false.

When x = 0, both sides are undefined (false).

When x > 0, the absolute values don't matter, so both sides are equal.

Therefore, answer is x < -1.