Is this inequality in $L^p(\Omega)$ true?

Question

Let $\Omega$ be an open bounded set and $u\in L^p(\Omega)$ with $p >1$. Let $\Omega^{\prime} = \left\lbrace x\in\Omega \mid u(x) >k\right\rbrace$ where $k$ denotes a positive constant such that $k>1$. It is true that $$\Vert u -k\Vert_{L^p(\Omega^{\prime})}\leq \Vert u\Vert_{L^p(\Omega^{\prime})}?$$ How could I proceed? Thank You in adcvance!