We define two new versions of integral and Frobenius closures of ideals which incorporate an auxiliary ideal and a real parameter. These additional ingredients are commonly used to adjust old definitions of ideal closures in order to generalize them to pairs. In the case of tight closure, similar generalizations exist due to N. Hara and K. I. Yoshida, as well as A. Vraciu. We study their basic properties and give computationally effective calculations of the adjusted tight, Frobenius, and integral closures in the case of affine semigroup rings in terms of the convex geometry of the associated exponent sets. Finally, we study submodules of the fraction field of a domain defined in terms of our adjusted closures and the application of the new closures to an F-nilpotent property for ideal pairs. This is a joint work with Kyle Maddox and Lance Miller.