Quotient of Lipschitz functions is lipschitz

Question

I wanna show that the quotient of two Lipschitz functions is Lipschitz. I feel like it should fall out with some kind of basic inequility argument but am struggling with the details. Anyone wanna help?

Answer 1

$f(x)\equiv 1$ and $g(x)=x$ are Lipschitz, but $f(x)/g(x)=1/x$ is not.