Let $||\cdot||$ denote the operator norm, with $T\in L(\mathbb{R}^n)$ and $S\in GL(n)$, and suppose that $T(\mathbf{x})=\mathbf{0}$. On one hand, we have that $$||S(\mathbf{x})||=||{S(\mathbf{x})-T(\mathbf{x})}||=||{(S-T)(\mathbf{x})}||\leq ||{S-T}||||{\mathbf{x}}||.$$ So, I need something like $$||\mathbf{x}||/||S^{-1}||\leq ||S(\mathbf{x})||$$ to obtain the result I want. I know that $1/||S^{-1}||\leq||S||$, but I don't see exactly how to use it.

EDIT: $\mathbf{x}\neq\mathbf{0}.$