Key Concepts of Inequalities

Inequalities are key in comparing expressions and understanding their relationships. They use symbols like < and > to show how values relate, and mastering them helps solve problems in algebra and real-life situations, like budgeting and resource management.

1. Definition of inequalities

   • An inequality is a mathematical statement that compares two expressions and shows the relationship between them.
   • It indicates that one expression is greater than, less than, or equal to another.
   • Inequalities can involve variables, constants, and mathematical operations.

2. Symbols used in inequalities (<, >, ≤, ≥)

   • The symbol "<" means "less than."
   • The symbol ">" means "greater than."
   • The symbol "≤" means "less than or equal to."
   • The symbol "≥" means "greater than or equal to."

3. Properties of inequalities

   • If you add or subtract the same number from both sides of an inequality, the inequality remains true.
   • If you multiply or divide both sides by a positive number, the inequality remains true; if by a negative number, the inequality sign reverses.
   • The transitive property states that if a < b and b < c, then a < c.

4. Solving linear inequalities

   • To solve a linear inequality, isolate the variable on one side using inverse operations.
   • Be mindful of the inequality sign when multiplying or dividing by negative numbers.
   • The solution set can be expressed in various forms, including interval notation.

5. Graphing inequalities on a number line

   • Use an open circle to represent values that are not included (for < or >).
   • Use a closed circle to represent values that are included (for ≤ or ≥).
   • Shade the region of the number line that satisfies the inequality.

6. Compound inequalities (AND, OR)

   • An "AND" compound inequality requires both conditions to be true simultaneously.
   • An "OR" compound inequality allows for either condition to be true.
   • Graphing compound inequalities involves shading the appropriate regions on the number line.

7. Absolute value inequalities

   • An absolute value inequality expresses the distance of a variable from a number.
   • It can be written as two separate inequalities: one for the positive case and one for the negative case.
   • Solutions can be represented as intervals on a number line.

8. Interval notation

   • Interval notation is a way to describe the set of solutions to an inequality using parentheses and brackets.
   • Parentheses indicate that an endpoint is not included, while brackets indicate inclusion.
   • It provides a concise way to express ranges of values.

9. Applications of inequalities in real-world problems

   • Inequalities can model constraints in various scenarios, such as budgeting, resource allocation, and optimization.
   • They help in determining feasible solutions within given limits.
   • Real-world problems often require interpreting inequalities to make informed decisions.

10. Inequalities with variables on both sides

    • To solve, first rearrange the inequality to get all variable terms on one side and constants on the other.
    • Be cautious of the inequality sign when performing operations.
    • The solution may lead to a range of values or a specific value depending on the inequality.