To simplify the expression

\[
\frac{2x - 1}{x - 1} - \frac{x - 2}{x - 1},
\]

we notice that both terms have a common denominator of \( x - 1 \). Therefore, we can combine the fractions:

\[
\frac{(2x - 1) - (x - 2)}{x - 1}.
\]

Now, simplify the numerator:

\[
2x - 1 - x + 2 = (2x - x) + (-1 + 2) = x + 1.
\]

Thus, the expression simplifies to:

\[
\frac{x + 1}{x - 1}.
\]

In conclusion, the simplified expression is

\[
\frac{x + 1}{x - 1}.
\]