Use AM-GM inequality$$px^q+\frac{q}{x^p}=\underbrace{x^q+x^q+\cdots+x^q}_{p\text{ times}}+\underbrace{\frac1{x^p}+\frac1{x^p}+\cdots+\frac1{x^p}}_{q\text{ times}} \ge p+q\\~\\$$ $$\implies px^q-p =qx^{-p}-q$$

Use AM-GM inequality$$px^q+\frac{q}{x^p}=\underbrace{x^q+x^q+\cdots+x^q}_{p\text{ times}}+\underbrace{\frac1{x^p}+\frac1{x^p}+\cdots+\frac1{x^p}}_{q\text{ times}} \ge p+q\\$$ $$\implies px^q-p =qx^{-p}-q$$

EDIT
This can be extended to the case where $p,q\in \mathbb{R^+}$. When $p,q\in \mathbb{R^+}$ we can use weighted AM-GM inequality

Use AM-GM inequality$$px^q+\frac{q}{x^p}=x^q+x^q+\cdots+x^q+\frac1{x^p}+\frac1{x^p}+\cdots+\frac1{x^p} \ge p+q$$ $$\implies px^q-p =qx^{-p}-q$$

Use AM-GM inequality$$px^q+\frac{q}{x^p}=\underbrace{x^q+x^q+\cdots+x^q}_{p\text{ times}}+\underbrace{\frac1{x^p}+\frac1{x^p}+\cdots+\frac1{x^p}}_{q\text{ times}} \ge p+q\\~\\$$ $$\implies px^q-p =qx^{-p}-q$$