I have a problem that boils down to two unknowns, $X_1$ and $X_2$, where:
$X_1 \cdot M + A\bmod N = X_2$
$X_1 \lt L_1\bmod N$
$X_2 \lt L_2\bmod N$
I can try every possible $X_1 \lt L_1$ until I hit one that produces an $X_2 \lt L_2\bmod N$ and solve one such problem in less than a minute. However, I have thousands of these to solve, so any increase in efficiency will greatly help.
Two questions I found indicate that inequality is meaningless in modulo / congruences. However, in this case, the inequality has a very specific meaning - to limit the range of valid values for $X_1$ and $X_2$. Those questions are: