Maximum Integer Value in Right Triangle

In triangle ABC, the lengths of sides AB, AC, and BC are represented by the variables $a$, $b$, and $c$, respectively. The lengths are related by the equation:

$$ a^2 + b^2 = c^2 $$

This indicates that triangle ABC is a right triangle. If $a$ is 9 units long, what is the maximum possible integer value of $b$ such that the triangle inequality is satisfied?