### Notes from Toppers

**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.

- A linear inequality in two variables is an inequality that can be expressed in the form

**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

- Linear inequalities can be used to solve a variety of real-world problems like:

**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

- 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:

**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.