Positive and negative roots of e to the power x equals to x.

Answer 1

e^x = x

since e^x is always positive, there are no negative roots.

Since the slope of y=x is 1, and the slope of y=e^x is greater than 1 for positive x, then since e^0>0, the two graphs never intersect.

In short, there are no real solutions.

Answer 2

Confirming Steve's conclusion:

https://www.wolframalpha.com/input/?i=Plot+y+%3D+e%5Ex,+y+%3D+x