# Inequalities

**Understanding Inequalities**

**Inequalities**: Are mathematical symbolic expressions implying that one quantity is less than or greater than another.**Viewing inequalities like equations**: Inequalities can be solved in a similar way to standard equations, but remember that when multiplying or dividing by a negative value, the sign of the inequality changes.

**Types of Inequalities**

**Linear Inequalities**: Simplest inequalities that are based on linear equations only. For example, 2x + 3 < 7.**Quadratic Inequalities**: Inequalities involving a quadratic expression, e.g., x^2 + 3x > 4.**Rational Inequalities**: Inequalities involving rational expressions or fractions. For example, (x + 1)/(x - 2) ≤ 0.-
**Absolute Value Inequalities**: Inequalities involving absolute values. E.g.,2x + 3 < 5.

**Techniques to Solve Inequalities**

**Isolation**: Aim to isolate the variable on one side of the inequality., similar to solving a normal equation.**Factorisation**: May be necessary to factorise before isolating the variable, especially for quadratic or other higher degree inequalities.**Sign Analysis**: Used to determine the solution interval by checking signs in intervals around critical points, often used in rational or absolute value inequalities.**Interval notation**: Used to represent the solution sets of inequality. Single solution intervals can be combined with union (∪) where the solution can be one interval or the other, and with intersection (∩) where the solution needs to fall in both intervals.

**Testing your Solution**

**Substitute back into the inequality**: Always check your solution for accuracy by substituting back into the original inequality.

**Graphical Interpretation of Inequalities**

**Number Line representation**: Inequalities are often represented on number lines, where open circles are used for greater than (‘>’) or less than (‘<’) values and filled circles for greater than or equal to (‘≥’) or less than or equal to (‘≤’) values.**XY-Plane representation**: For two-variable inequalities, half-plane notions are used, where the area of the plane above or below the line is shaded to show the solution space.