We propose an explanation of the island and banana states and the relation between atoll and Z-track sources, constituting a unified model for low-mass X-ray binaries (LMXB). We find a dramatic transition at a luminosity of above which the high-energy cut-off of the Comptonized emission in all sources is low at a few keV. There is thermal equilibrium between the neutron star at ∼2 keV and the Comptonizing accretion disc corona (ADC) causing the low in the banana state of atolls and all states of the Z-track sources. Below this luminosity, increases towards 100 keV causing the hardness of the island state. Thermal equilibrium is lost, the ADC becoming much hotter than the neutron star via an additional coronal heating mechanism. This suggests a unified model of LMXB: the banana state is a basic state with the mass accretion rate M increasing, corresponding to the normal branch of Z-track sources. The island state has high ADC temperature, this state not existing in the Z-sources with luminosities much greater than the critical value. The Z-track sources have an additional flaring branch consistent with unstable nuclear burning on the neutron star at high M˙. This burning regime does not exist at low M˙ so this branch is not seen in atolls (except GX atolls). The horizontal branch in Z-track sources has a strong increase in radiation pressure disrupting the inner disc and launching relativistic jets.