Systems and sets of inequalities; exponential and logarithmic systems
Tell "and" (a system, the common part) from "or" (a set, the union), solve mixed systems and systems of exponential and logarithmic equations.
📖 Start here: read the steps of the “Lesson” in order (1, 2, 3 …) and answer each question. Then continue with “Practice”.
- Russia: grade 11
Guide
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: ax·ay = ax + 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.