Calculus: Is $(x-1)/f(x)$ decreasing?

Suppose $f$ is a continuous function and $f(x) \geq x-1$ for all $x>1$. It is also given that $f(x)$ is a monotonically increasing function in $x>1$. Can I say that$$\frac{x-1}{f(x)}\quad\text{is an monotonically decreasing function in $x>1$ ?}$$