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An illustration of Bernoulli's inequality, with the graphs of $y=(1+x)^{r}$ and $y=1+rx$ shown in red and blue respectively. Here, $r=3.$

In mathematics, Bernoulli's inequality (named after Jacob Bernoulli) is an inequality that approximates exponentiations of $1+x$ . It is often employed in real analysis. It has several useful variants:^[1]

^ Brannan, D. A. (2006). A First Course in Mathematical Analysis. Cambridge University Press. p. 20. ISBN 9781139458955.