Showing that solutions to an ODE does not escape to $\infty$ in finite time I have an ODE

$$x'=f(x)$$

and that there are constants $C$, $K$ such that $|f(x)| ≤ C|x| + K$ for all x.

How can I show that the solutions to this ODE does not escape to infinity in finite time?

I tried to write it as $\frac{|x'|-K}{|x|} ≤ C$ but don't think I could integrate this.
You can
 bound $|x(t)|$ by a solution of the linear equation $y' = C y + K$.
Hint: Use the mean value theorem: $\mid x(u)-x(v)\mid \leq \mid x'(c)(u-v)\mid =\mid f(c)(u-v)\mid \leq (C\mid c\mid+K)(u-v)$ where $c\in [u,v]$.