Linear Inequalities

• Introduction & Basic Concepts

  • Algebraic Statements in Mathematics
  • Inequality Symbols: <, >, <=, >=
  • Difference between Equations and Inequalities
  • Strict Inequalities: < and >
  • Slack Inequalities: <= and >=

• Algebraic Solutions in One Variable

  • Definition of Solution
  • Replacement Set vs Solution Set
  • Rules for Solving
    • Addition/Subtraction of equal numbers
    • Multiplication/Division by positive numbers (sign remains unchanged)
    • Multiplication/Division by negative numbers (reversing the inequality sign)
  • Representation of Solutions
    • Number line representation (solid dot vs hollow dot)
    • Interval notation [a, b], (a, b), [a, b), (a, b]

• Graphical Solutions in Two Variables

  • Linear Inequality in Two Variables: $ax + by < c$ (or $\le, >, \ge$)
  • Steps to Solve Graphically
    • Replace inequality sign with equality to form boundary line ($ax + by = c$)
    • Draw boundary line (dashed line for strict, solid line for slack)
    • Choose a test point (commonly origin $(0,0)$)
    • Shade the half-plane containing the solution
  • Non-strict inequalities include the boundary line

• Systems of Linear Inequalities

  • Solving multiple inequalities simultaneously in two variables
  • Finding the Solution Region (Common shaded area)
  • Intersection of half-planes
  • Unbounded and Bounded solution regions (Introduction to Linear Programming context)