Prove inequality with multiple variables in a field

Question

I know: $ p1\ge 0 , p2\ge0$ and $p1+p2=1$ as well as $x\ge y$. How do I prove the following inequality?

$$x\ge p1*x+p2*y$$

Answer 1

Multiply both sides of $x\ge y$ by $p2$, use $p2=1-p1$, then move the $p1$ term to the other side. $$x\ge y\\ p2*x\ge p2*y\\ (1-p1)*x\ge p2*y\\ x-p1*x\ge p2*y\\ x\ge p1*x+p2*y $$

Answer 2

the given inequality is equivalent to $$(x-y)(1-p_1)\geq 0$$ and this is true, since $$x\geq y$$ and $$1-p_1=p_2\geq 0$$