Linear Inequality in Two Variables - Detailed Notes

1. Linear Equations and Inequalities:

• NCERT Reference:
  • Class 11: Chapter 4 (Linear Equations)
• Key Points:
• A linear equation is an algebraic equation of degree 1.
• A linear inequality is an algebraic inequality of degree 1.

2. Linear Inequality in Two Variables:

• NCERT Reference:
  • Class 11: Chapter 6 (Linear Inequalities)
• Key Points:
  • A linear inequality in two variables is an inequality that can be expressed in the form Ax + By + C >/< 0, where A, B, and C are real numbers and x and y are variables.
  • Linear inequalities in two variables can be represented graphically as lines or half-planes.

3. Graphing Linear Inequalities:

• NCERT Reference:
  • Class 11: Chapter 6 (Linear Inequalities)
• Key Points:
• To graph a linear inequality, first, need to find the boundary line by setting the inequality equal to zero.
• Then, determine which side of the boundary line to shade.
• The solution region is the shaded half-plane that satisfies the inequality.

4. Intersection and Union of Linear Inequalities:

• NCERT Reference:
• Class 11: Chapter 6 (Linear Inequalities)
• Key Points:
  • The intersection of two or more linear inequalities is the region that satisfies all the inequalities.
• The union of two or more linear inequalities is the region that satisfies at least one of the inequalities.

5. Applications of Linear Inequalities:

• NCERT Reference:
  • Class 11: Chapter 6 (Linear Inequalities)
• Class 12: Chapter 12 (Linear Programming)
• Key Points:
  • Linear inequalities can be used to solve a variety of real-world problems like:
    • Maximization or minimization of objective functions under given constraints
    • Break-even analysis
    • Resource allocation
    • Optimization problems

6. System of Linear Inequalities:

• NCERT Reference:
  • Class 11: Chapter 6 (Linear Inequalities)
• Key Points:
• A system of linear inequalities is a set of two or more linear inequalities.
• To solve a system of linear inequalities, graph each inequality and find the region that satisfies all the inequalities.

7. Linear Programming:

• NCERT Reference:
  • Class 12: Chapter 12 (Linear Programming)
• Key Points:
  • Linear programming is a method for solving optimization problems with linear objective functions and linear constraints. There are two main methods for solving linear programming problems:
    • Graphical method
    • Simplex method

8. Applications in Optimization:

• NCERT Reference:
• Class 12: Chapter 12 (Linear Programming)
• Key Points:
• Linear inequalities can be used to model and solve optimization problems in various fields like:
  • Engineering
  • Economics
  • Business
  • Finance

9. Word Problems:

• NCERT Reference:
  • Class 11: Chapter 6 (Linear Inequalities)
  • Class 12: Chapter 12 (Linear Programming)
• Key Points:
• Many real-world problems can be modeled using linear inequalities.
• To solve these problems, identify the variables involved, set up the linear inequalities, and solve them graphically or using other methods.

10. Advanced Concepts:

• Linear inequalities with absolute values:
  • NCERT Reference: Class 12: Chapter 1 (Relations and Functions)
• Linear inequalities with quadratic or exponential functions:
  • NCERT Reference: Class 11: Chapter 8 (Quadratic Equations), Chapter 13 (Limits and Derivatives), Class 12: Chapter 9 (Applications of Derivatives)

Remember to practice solving a variety of problems related to each subtopic to solidify your understanding and confidently tackle linear inequalities for the JEE exam.