inequalities for differences of powers

Oftentimes, one needs to estimate differences of powers of real numbers.The following inequalities are useful for this purpose:

$$nx\le {(1+x)}^n-1$$

$${(1+x)}^n-1\le \frac{nx}{1-(n-1)x}$$

Here $n$ is an integer greater than 1.The first inequality holds when , the second inequality holdswhen , and the third inequality holds when .Equality can only occur in the latter two inequalities when $x=0$.