ELMO SL 2014 A2

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that $$p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c).$$ Proposed by AJ Dennis