Linear Inequalities – Class 11 Maths (NCERT Based)
The chapter Linear Inequalities introduces methods to solve inequalities involving algebraic expressions and represents their solutions on a number line. This chapter is essential for understanding constraints in real-life problems and for higher-level topics in mathematics.
This content strictly follows the NCERT Class 11 Maths syllabus and is explained in simple, student-friendly language.
📖 What is a Linear Inequality?
A linear inequality is similar to a linear equation, but instead of an equal sign (=), it uses one of the following symbols:
- < (less than)
- > (greater than)
- ≤ (less than or equal to)
- ≥ (greater than or equal to)
Example:
2x + 5 ≤ 9
🔹 Types of Linear Inequalities
- Linear inequalities in one variable
- Example: 3x − 7 > 2
- Linear inequalities in two variables
- Example: x + y ≤ 5
✍️ Solving Linear Inequalities
1. In One Variable
- Solve like an equation, but reverse the inequality if multiplying or dividing by a negative number.
- Represent the solution on a number line.
Example:
−2x + 3 < 7
Step 1: −2x < 4
Step 2: x > −2 (inequality reversed)
2. In Two Variables
- Represent inequalities in xy-plane.
- Graph the boundary line first:
- Solid line for ≤ or ≥
- Dashed line for < or >
- Shade the region satisfying the inequality.
- Check a test point to confirm the solution region.
Example:
x + y ≤ 4 → Graph the line x + y = 4 and shade below the line.
🔹 Important Rules for Linear Inequalities
- Adding or subtracting the same number on both sides does not change the inequality.
- Multiplying or dividing by a positive number keeps the inequality same.
- Multiplying or dividing by a negative number reverses the inequality.
- Inequalities can be combined using “and” / “or” statements.