Chapter Summary: Linear Inequalities

  • One Variable: An inequality in one variable can be solved using algebraic rules. The most important rule is to reverse the inequality sign whenever you multiply or divide both sides by a negative number.

  • Two Variables: A linear inequality in two variables (e.g., ax+by>cax+by > c) represents a half-plane in the Cartesian coordinate system. The solution is found by drawing the boundary line and shading the correct half-plane determined by a test point.

  • Systems of Inequalities: The solution to a system of two or more inequalities is the intersection of their individual solution regions. This common area is known as the feasible region.

🎯 Strategic Tips for JEE Main & Advanced

  • Master the Wavy Curve Method: For non-linear inequalities like (x−1)(x+2)(x−3)>0\frac{(x-1)(x+2)}{(x-3)} > 0, do not cross-multiply. Use the wavy curve (or sign analysis) method:
  1. Find all critical points (where factors equal zero).
  2. Place them on a number line.
  3. Check the sign of the expression in each interval to find the solution.

  • Memorize Modulus Inequality Rules: These are essential for solving inequalities involving absolute values quickly.

  • ∣x∣<a  ⟺  −a<x<a|x| < a \iff -a < x < a

  • ∣x∣>a  ⟺  x<−a or x>a|x| > a \iff x < -a \text{ or } x > a

  • Foundation for Linear Programming: The graphical method for solving systems of inequalities is the fundamental skill for Linear Programming Problems (LPP), a key topic in Class 12. Focus on accurately identifying the feasible region and its corner points (vertices).

  • Avoid Common Mistakes: Never multiply or divide an inequality by a variable or an expression unless you are certain of its sign. For example, to solve x−1x>0\frac{x-1}{x} > 0, you cannot multiply by xx. You must use the wavy curve method.