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.