The determination of the low-energy cutoff to nonthermal electron distributions is critical to the calculation of the nonthermal energy in solar flares. The most direct evidence for low-energy cutoffs is flattening of the power-law, nonthermal X-ray spectra at low energies. However, because of the plasma preheating often seen in flares, the thermal emissions at low energies may hide such spectral flattening of the nonthermal component. We select a category of flares, which we call ``early impulsive flares,'' in which the >25 keV hard X-ray (HXR) flux increase is delayed by less than 30 s after the flux increase at lower energies. Thus, the plasma preheating in these flares is minimal, so the nonthermal spectrum can be determined to lower energies than in flares with significant preheating. Out of a sample of 33 early impulsive flares observed by the Ramaty High Energy Solar Spectroscopy Imager (RHESSI), nine showed spectral flattening toward low energies. In these events, the break energy of the double power-law fit to the HXR spectra lies in the range 10-50 keV, significantly lower than the value we have seen for other flares that do not show such early impulsive emissions. In particular, it correlates with the HXR flux. After correcting the spatially integrated spectra for albedo from isotropically emitted X-rays and using RHESSI imaging spectroscopy to exclude the extended albedo halo, we find that albedo associated with isotropic or nearly isotropic electrons can only account for the spectral flattening in three flares near Sun center. The spectral flattening in the remaining six flares is found to be consistent with the existence of a low-energy cutoff in the electron spectrum, falling in the range 15-50 keV, which also correlates with the HXR flux.