Assume that for a given minimization problem with only integer solutions, it is $NP$-hard to decide if the optimal solution is 5 or 6. I.e., a polynomial-time algorithm with an approximation ratio better than 6/5 would imply $P=NP$.
1) Does this imply that the problem is $APX$-hard as well?
2) Is there a common way of stating this inapproximability fact, besides stating that "it is $NP$-hard to approximate with an approximation ratio strictly better than 6/5"?
Thank you!