Is the following statement true or false?

For every odd integer $n,|$~$\frac{n}{2}$~$|=\frac{n+1}{2}$

To answer this question, let $n$ be any odd integer. By definition of odd, $n=2k+1$ for some integer $k,$ and so$,$ by substitution, $|$~$\frac{n}{2}$~$|=|$~$\frac{2k+1}{2}$~$|.$ starting with this equality, express $|$~$\frac{n}{2}$~$|$ in temes of $k$ as simply as possible without using the celling notation. The result is

$|$~$\frac{n}{2}$~$|=|$~$|$

Also, when $n+1$ is expressed in terms of $k$ and is completely simplified, the result is