Systems and sets of inequalities; exponential and logarithmic systems

In the Russian programme systems and sets of inequalities are met in grades 9-11; systems of exponential and logarithmic equations in grade 11 (in detail at the advanced level, simple cases at the basic one).

A system of inequalities (a curly brace) needs all conditions at once: the solution is the intersection. A set (a square bracket) needs at least one: the solution is the union.

In a mixed system find the solutions of each part first, then take the common part. The answer is often a union of intervals with the sign ∪.

In a system of exponential equations use the laws of powers: a^x·a^y = a^{x + y}; bring both sides to one base and pass to a system for the exponents.

In a system of logarithmic equations pass to an equation for the arguments and always check the domain: the argument of a logarithm must be positive.