Modulus proprierty in e power to the Real part of a complex number

Question

I am studying Complex Analysis with the book Schaum's Complex Variables - from Bailey. See the question bellow and the solution for the alternative b.

There is a point that I do not understand in this proof.

Why |$e^x$ | turns to $e^x$ ?

If x is the Re(z) (the real part of z), |$e^x$ | can be $e^x$ or $e^{-x}$ depending if x is positive or negative.

What error am I making here? Am I using some wrong assumption?

Thanks.

Answer 1

Since $x\in\mathbb R$, $e^x>0$. Therefore, $|e^x|=e^x$.