Bernstein inequalities and to Hoeffding's inequality. Economic inequalities exist between different ethnic groups. Research shows that segregation along racial lines contributes to public goods inequalities. The inferences involved reduce themselves to solving systems of linear inequalities. The U.S. is one of the countries that have noticeable racial inequalities. The field of Diophantine approximation deals with the cases of "Diophantine inequalities". Adam Smith addressed the question of whether vast inequalities of wealth represented progress. This implies taking into account historical and current inequalities among groups. Possession of livestock allowed competition between households and resulted in inherited inequalities of wealth. The materialist approach offers insight into the sources of health inequalities among individuals and nations. There are great inequalities in health between states. There is considerable research into inequalities in health care. This can be proved, e.g., by the norm inequalities. It also has one of the highest income inequalities among developed countries. Some may involve great inequalities of income.