jmc

Setting the mean of the set's members equal to $x-4.5$, we get the equation $$\frac{5+8+10+18+19+28+30+x}{8}=x-\mathrm{4.5.}$$Simplifying the left-hand side, we get $$\frac{118+x}{8}=x-\mathrm{4.5.}$$Multiplying by $8$ gives $118+x=8x-36$. Then $7x=118+36=154$. Thus $x=\frac{154}{7}=\boxed{22}.$