I am working on a problem where I am dealing with $$P(-Z \le x)$$ where Z is a standard normal random variable. I am trying to figure out how to get to a cdf from here but I am not sure if I am using the correct logic. I write that$$P(-Z \le x) = P(Z \ge -x) = 1 - P(Z \lt -x)$$
Can I then state that this is equal to $$1 - Fz(-x)?$$
The fact that it is no longer less than or equal to in the above line is the reason I think I may have gone wrong.
Any feedback is appreciated.
In general, $P(Z<-x)=\lim_{z \to -x^-} F_Z(z)$, but since $Z$ is continuously distributed, $F_Z$ is a continuous function, so this is just $F_Z(-x)$.