Finding parameter range of inequation with no solution

Question

Let the solution set of the following inequation

$$6 k - 2 x + k x^2 < 0, \quad k \neq 0$$ be an empty set. How can I find find the range of parameter $k$?

The following code gives an answer:

Reduce[{Discriminant[k x^2 - 2 x + 6 k, x] <= 0, k > 0}, k]
(* k >= 1/Sqrt[6] *)

However, the code below gives an error message Resolve::elemc. Can I fix the code somehow?

ForAll[x, x ∈ {}, k x^2 - 2 x + 6 k < 0 && k != 0]
Resolve[%, Reals]
Reduce[%, k, Reals]
(* During evaluation of In[540]:= Resolve::elemc: Unable to resolve 
   the domain or region membership condition x∈{}.
(* During evaluation of In[540]:= Resolve::elemc: Unable to resolve
   the domain or region membership condition `LocalVariable$1∈{}.
(* During evaluation of In[540]:= Resolve::elemc: Unable to resolve 
   the domain or region membership condition x∈{}.
(* ... *)

Answer 1

Let us consider the opposite inequality k x^2 - 2 x + 6 >= 0 and write down your preposition as

ForAll[x,k x^2 - 2 x + 6 >= 0]

Then

Resolve[ForAll[x, k x^2 - 2 x + 6 >= 0], Reals]

k >= 1/6

As we see k!=0.