About f(x)+g(x)<=c; f(x) is a convex function; g(x) is a concave function; and c is constant;

how to transform this non-convex to convex, is slack variable can be used for solving, such as

g(x)>=h(x) and h(x) is convex or affine, then f(x)+h(x)<=c is convex?

You need to be able to show that f(x)+g(x) is altogether convex, otherwise it is not a convex problem.