Expected value on non-normalised PDFs

Question

Suppose the following is known: $$\int{g(x)dx}>\int{g'(x)dx}$$

Considering that $g=kf$ and $g'=k'f'$ where $f$ and $f$ are probability distributions on $X\in[0,1]$. Is the following true:

$$E_g(x)> E_{g'}(x)$$

More rigorously:

$$\int{xg(x)dx}>\int{xg'(x)dx}$$

Answer 1

That is not true. Simple example: $g(x)=\frac{7}{12}$ and $g'(x)=x$.