We use new ALMA observations to investigate the connection between dense gas fraction, star formation rate (SFR), and local environment across the inner region of four local galaxies showing a wide range of molecular gas depletion times. We map HCN(1-0), HCO^+^(1-0), CS(2-1), ^13^CO(1-0), and C^18^O(1-0) across the inner few kiloparsecs of each target. We combine these data with short-spacing information from the IRAM large program EMPIRE, archival CO maps, tracers of stellar structure and recent star formation, and recent HCN surveys by Bigiel+ (2016ApJ...822L..26B) and Usero+ (2015AJ....150..115U). We test the degree to which changes in the dense gas fraction drive changes in the SFR. I_HCN_/I_CO_ (tracing the dense gas fraction) correlates strongly with I_CO_ (tracing molecular gas surface density), stellar surface density, and dynamical equilibrium pressure, P_DE_ (Elmegreen 1989ApJ...338..178E). Therefore, I_HCN_/I_CO_ becomes very low and HCN becomes very faint at large galactocentric radii, where ratios as low as I_HCN_/I_CO_~0.01 become common. The apparent ability of dense gas to form stars, {Sigma}_SFR_/{Sigma}_dense_ (where {Sigma}_dense_ is traced by the HCN intensity and the star formation rate is traced by a combination of H{alpha} and 24{mu}m emission), also depends on environment. {Sigma}_SFR_/{Sigma}_dense_ decreases in regions of high gas surface density, high stellar surface density, and high P_DE_. Statistically, these correlations between environment and both {Sigma}_SFR_/{Sigma}_dense_ and I_HCN_/I_CO_ are stronger than that between apparent dense gas fraction (I_HCN_/I_CO_) and the apparent molecular gas star formation efficiency {Sigma}_SFR_/{Sigma}_mol_. We show that these results are not specific to HCN.