Why does one of the following constraints define a convex set while another defines a non-convex set?

Question

I have three constraints

$$x \leq \frac{gb}{y+a} \tag{Constraint 1}$$

$$g \leq G \tag{Constraint 2}$$

$$x \leq \frac{Gb}{y+a} \tag{Constraint 3}$$

where $G,y,a,b > 0$ and $x \geq 0$. I want to know why constraint 1 defines a non-convex set while constraint 3 defines a convex set? Any help in this regard will be much appreciated. Thank you.

Answer 1

Assuming that the only variables in (C1) are $x$ and $g$ and the other quantities are positive, the resulting set is a convex halfspace. The way you wrote this down you are mixing variables and constants, but it is crucial to know which numbers are fixed to accurately answer this question.