Finding the set {$z\in \mathbb{C}|2z+2i-3\in [0,\infty)$}

Question

This is how I tried this but not really sure:

{$z\in \mathbb{C}|2z+2i-3\in [0,\infty)$}$\implies${$z\in \mathbb{C}|2z+2i\in [3,\infty)$}$\implies${$z\in \mathbb{C}|z+i\in [3/2,\infty)$}.

Now if $z=x+iy$ then $z+i=x+i(y+1)\in [3/2,\infty)$$\implies$$y=-1,x\in [3/2,\infty)$. So,

{$z\in \mathbb{C}|2z+2i-3\in [0,\infty)$}$=${$z=x+iy\in \mathbb{C}|y=-1,x\in [3/2,\infty)$}

Is this correct?