1. The problem is to simplify the expression $|x-1| + 1$. 2. Recall that the absolute value function $|x-1|$ outputs the distance between $x$ and $1$ regardless of direction. It is always non-negative. 3. Since $|x-1|$ is already simplified (it cannot be simplified algebraically without knowing if $x$ is less than or greater than 1), the expression $|x-1| + 1$ is in its simplest form. 4. The expression always outputs a value that is $1$ greater than the distance between $x$ and $1$ on the number line. Therefore, the simplified expression is $|x-1| + 1$ itself.