Jensen's inequality with affine combination

From page 217 of 'Convex functions' by Arthur Wayne Roberts, Dale Varberg (exercise F) i want proof that: $f:\mathbb{R}\longrightarrow\mathbb{R}$ is affine iff

$$ f\biggl(\displaystyle\sum\limits_{i=1}^n\lambda_ix_i\biggr)\le \displaystyle\sum\limits_{i=1}^n\lambda_if(x_i)$$

for all $\lambda_1,...,\lambda_n,x_1,...,x_n\in\mathbb{R}$ with $\displaystyle\sum\limits_{i=1}^n\lambda_i=1$.

If $f$ is affine (i.e. $f(x)=mx+q$ with $m,q\in\mathbb{R}$) is obvious that $ f\biggl(\displaystyle\sum\limits_{i=1}^n\lambda_ix_i\biggr)= \displaystyle\sum\limits_{i=1}^n\lambda_if(x_i)$ but why conversely?