Yes, you can select any value in the determined interval. For the last variable that is left, we have already such an interval determined by the greatest of the lower bounds and the least of the upper bounds.
Once determined such a value, you get another interval in the same way for the next variable, and so on. In particular, you can thereby select integer values if there is an integer solution which makes Fourier-Motzkin not more difficult for integers than for reals. Even more, you can also handle inequalities with < instead of <= which is not possible with Simplex.