I am working on a problem where I want to find the distribution of $X$.
$$P(X \le x) = P(Z \le x, U \le \frac{1}{2}) + P(-Z \le x, U \gt \frac{1}{2})$$
Where $Z$~$N(0,1)$, $U$~$(0,1)$ and $Z$ and $U$ are independent. By this independence I get $$=P(Z \le x) P( U \le \frac{1}{2}) + P(-Z \le x)P( U \gt \frac{1}{2})$$ $$=F_z(x)F_u(\frac{1}{2}) + (1 - F_z(-x))(1-F_u(\frac{1}{2}))$$ $$=F_z(x)F_u(\frac{1}{2}) + 1 - F_z(-x) - F_u(\frac{1}{2}) + F_z(-x)F_u(\frac{1}{2})$$
But now I have no idea how to handle the $F_z(-x)$. Any feedback would be appreciated.
The distribution of $Z$ is symmetric so $P(-Z \leq x)$ is same as $P(Z \leq x)$. Now you can complete the argument. You will see that $X$ also has the standard normal distribution.