$$X(\omega) = \frac12-j\omega + \frac12+j\omega$$
Without feedback, a student can spend 10 hours on one problem only to discover they violated a positive-definiteness assumption on page three. The provides that feedback loop, validating your approach or revealing the elegant shortcut you missed.
To illustrate the importance of mathematical methods and algorithms in signal processing, let's consider a few examples from a solution manual.