The coupling of the compressional instabilities in a two-component, hot and cold inhomogeneous plasma to the resonant shear Alfv¿n mode is studied in a slab geometry with straight field lines. The lowest order effect of the field inhomogeneity is incorporated into the dispersion relation by retaining the ∇B drift of the particles. It is found that the high-phase velocity compressional mode with &ohgr;/k∥>V∥i shows insignificant coupling to the shear Alfv¿n wave, while the low-phase velocity branch with &ohgr;/k∥<V∥i shows significant coupling. It is also shown that the coupling factor is of the form &Bgr;∥, nc/nik⊥ 2rR2, so that a more realistic field geometry should be used to model the magnetospheric field in the study of low-frequency geomagnetic pulsations.