On the positivity of the positive part of a closed symmetric operator?

Question

Let $A$ be a closed symmetric densely defined unbounded operator (which is not self-adjoint). Is it true that $|A|+A\geq0$? (Where $|A|=\sqrt{A^*A}$), i.e. whether it holds that $$\langle(|A|+A)x,x\rangle\geq 0,~\forall x\in D(A)?$$

Many thanks.

Math