1. **Problem Statement:** Explain the concept of inequalities in mathematics. 2. **Definition:** An inequality compares two values or expressions to show if one is less than, greater than, less than or equal to, or greater than or equal to the other. 3. **Types of Inequalities:** The main types are: - Strict inequalities: $a < b$ or $a > b$ mean $a$ is strictly less or greater than $b$. - Non-strict inequalities: $a \leq b$ or $a \geq b$ mean $a$ is less than or equal to, or greater than or equal to $b$, respectively. 4. **Properties:** - If $a < b$, then $a + c < b + c$ for any $c$ (adding the same number preserves inequality). - If $a < b$ and $c > 0$, then $ac < bc$ (multiplying by positive maintains inequality). - If $a < b$ and $c < 0$, then $ac > bc$ (multiplying by negative reverses inequality). 5. **Solving Inequalities:** Similar to equations, but when multiplying or dividing by a negative number, reverse the inequality sign. 6. **Graphing:** Inequalities can be represented on number lines or coordinate planes, shading the region where the inequality holds true. 7. **Example:** Solve $2x - 3 > 5$. Step 1: Add 3 to both sides: $2x > 8$ Step 2: Divide both sides by 2 (positive number, so inequality sign unchanged): $x > 4$ 8. **Conclusion:** Inequalities express relationships of order and can be manipulated carefully to find solution sets.