Probability density function of $\log(1-|X|/2)$ where $X$ is a uniformly distributed variable

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I'm a bit confused on how to solve this. This is my work so far:

problem

But the last step I have done is apparently wrong. Any help would be appreciated!

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$\mathsf P(\lvert X\rvert > n)=\mathsf P(X>n \;\color{blue}\bigcup\; X<-n)$ for any $n>0$

That is: OR not AND.