Example 1 Solve the inequalities: (1) $\sqrt{x-1}<1$; (2) $\sqrt{2 x-3} \leqslant \sqrt{x-1}$.
Solution (1) Since $\sqrt{x-1}<1 \Leftrightarrow\left\{\begin{array}{l}x-1 \geqslant 0, \\ x-1<1 .\end{array}\right.$
Therefore, the solution set of the original inequality is $[1,2)$.
(2) Since $\sqrt{2 x-3} \leqslant \sqrt{x-1} \Leftrightarrow\left\{\begin{array}{l}2 x-3 \geqslant 0, \\ x-1 \geqslant 0, \\ 2 x-3 \leqslant x-1 .\end{array}\right.$

Therefore, the solution set of the original inequality is $\left[\frac{3}{2}, 2\right]$.
Solution notes When solving irrational inequalities, it is essential to pay attention to the changes in the range of values for the unknown after transformation, and to ensure the equivalence of transformations.