Proof $x$, $1+nx≤ (1+x)^n$ Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$. 

Don't have a idea to start this one. I don't know how to use math induction yet, so I need a answer without that.
Using the binomial theorem:

$$(1+x)^n=1+nx+\dots$$

Where $\dots$ is non-negative, as it is a polynomial in $x$ with non-negative coefficients and $x\geq 0$.
Equality holds for $x=1$. Then, taking the derivative,

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