In this booklet the author did not pursue the aim of presenting the basic properties of inequalities and made an attempt only to
familiarize students of senior classes with some particularly
remarkable inequalities playing an important role in various sections of higher mathematics and with their use for finding the greatest and the least values of quantities and for calculating some limits. The book contains 63 problems, 35 of which are provided with detailed solutions, composing thus its main subject, and 28 others are given in Sections 1.1 and 2.1, 2.3, 2.4 as exercises for individual training. At the end of the book the reader will find the solutions to the given exercises.