I'm having trouble understanding how the CDF is found in the solution below:
We can assume the units are chosen so that the stick has length $1$. Let $L$ be the length of the longer piece, and let the break point be $U \sim Unif(0,1)$. For any $l \in [1/2,1]$, observe that $L<l$ is equivalent to $U<l,1-U<l$, which can be written as $1-l<U<l$. We can thus obtain $L$'s CDF as $$F_L(l) = P(L<l)=P(1-l<U<l)=2l-1$$
Can someone please explain why $L<l$ is equivalent to $U<l,1-U<l$? Isn't the break point $U$ in between $[1/2,1]$?