Solving linear functional inequalities with a computer

We describe a universal procedure that allows to obtain continuous solutions of inequalities of the form $$\begin{aligned} a_1f(\alpha _1x+(1-\alpha _1)y)+\cdots +a_nf(\alpha _nx+(1-\alpha _n)y)\le 0,\;x,y\in I, \end{aligned}$$ where $$I\subset {\mathbb {R}}$$ is an interval, $$f:I\rightarrow {\mathbb {R}}$$ is an unknown function and $$a_i\in {\mathbb {R}},$$ $$\alpha _i\in [0,1]$$ are given numbers. After presenting and explaining all necessary theoretical details, we provide a computer program that solves inequalities of such type.