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Understanding Inequalities and Their Properties

Introduction to Inequalities

• The video introduces the concept of inequalities, defined as a relationship of inequality between two algebraic expressions involving one or more variables.

• An example provided is x - 3 geq 10 + x , illustrating the use of inequality signs such as greater than, less than, greater than or equal to, and less than or equal to.

Solving Inequalities

• To solve an inequality means finding all values of the variable that satisfy the inequality condition. For instance, x > 2 includes all numbers greater than 2.

• The solution set for x > 2 includes numbers like 3, 4, 5, etc., extending infinitely in the positive direction.

Solution Representation

• Solutions can be represented in various forms: interval notation, graphical representation, and set notation.

• The process involves isolating the variable (commonly x ), similar to solving algebraic equations.

Example of Solving an Inequality

• An example given is x + 7 < 12 . By moving '7' across the inequality sign, it becomes x < 12 - 7 , simplifying to x < 5 .

Properties of Inequalities

• Key properties include:

• First Property: Adding or subtracting the same number on both sides does not change the direction of the inequality.

• Second Property: Multiplying or dividing by a positive number keeps the direction unchanged.

• Third Property: Multiplying or dividing by a negative number reverses the direction of the inequality.

Types of Inequalities

• Different types discussed include linear inequalities, double inequalities, quadratic inequalities, and absolute value inequalities. This concludes an overview of inequalities and their properties.