We present constraints on variations in the initial mass function (IMF) of nine local early-type galaxies based on their low mass X-ray binary (LMXB) populations. Comprised of accreting black holes and neutron stars, these LMXBs can be used to constrain the important high mass end of the IMF. We consider the LMXB populations beyond the cores of the galaxies ($>0.2R_{e}$; covering $75-90%$ of their stellar light) and find no evidence for systematic variations of the IMF with velocity dispersion ($sigma$). We reject IMFs which become increasingly bottom heavy with $sigma$, up to steep power-laws (exponent, $alpha>2.8$) in massive galaxies ($sigma>300$km/s), for galactocentric radii $>1/4 R_{e}$. Previously proposed IMFs that become increasingly bottom heavy with $sigma$ are consistent with these data if only the number of low mass stars $(<0.5M_{odot}$) varies. We note that our results are consistent with some recent work which proposes that extreme IMFs are only present in the central regions of these galaxies. We also consider IMFs that become increasingly top-heavy with $sigma$, resulting in significantly more LMXBs. Such a model is consistent with these observations, but additional data are required to significantly distinguish between this and an invariant IMF. For six of these galaxies, we directly compare with published IMF mismatch parameters from the Atlas3D survey, $alpha_{dyn}$. We find good agreement with the LMXB population if galaxies with higher $alpha_{dyn}$ have more top-heavy IMFs -- although we caution that our sample is quite small. Future LMXB observations can provide further insights into the origin of $alpha_{dyn}$ variations.
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