We use new ALMA observations to investigate the connection between dense gas fraction, star formation rate, 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 kpc 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 et al. and Usero et al. 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, $P_{DE}$. Therefore, $I_{HCN}/I_{CO}$ becomes very low and HCN becomes very faint at large galactocentric radii, where ratios as low as $I_{HCN}/I_{CO} sim 0.01$ 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.
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