Chapter 5: Linear Inequalities

Summary

• Linear inequalities involve expressions with inequality signs: <, >, ≤, ≥.
• They can be solved for different sets of numbers: natural numbers, integers, and real numbers.
• Graphical representation of solutions is essential for understanding.
• Rules for solving inequalities include adding/subtracting equal numbers and multiplying/dividing by positive numbers without changing the inequality sign, but reversing it when multiplying/dividing by negative numbers.
• Examples include solving inequalities for average marks, costs, and temperature ranges.

Key Examples

• Example 1: Solve 30x < 200 for natural numbers: {1, 2, 3, 4, 5, 6}.
• Example 2: Solve 5x - 3 < 3x + 1 for integers: {..., -4, -3, -2, -1, 0, 1}.
• Example 3: Solve 4x + 3 < 6x + 7: x > -2.
• Example 4: Solve 7x + 3 < 5x + 9: x < 3.

Important Diagrams

• Number Line Representation:
  • Solid dot indicates included values.
  • Open circle indicates excluded values.
  • Arrows indicate ranges extending to infinity.

Common Mistakes & Exam Tips

• Mistake: Forgetting to reverse the inequality sign when multiplying/dividing by a negative number.
• Tip: Always check the solution by substituting values back into the original inequality.

Chapter 5: Linear Inequalities

5.1 Introduction

• Study of linear inequalities in one and two variables.
• Useful in various fields: science, mathematics, statistics, economics, psychology.

5.2 Inequalities

Definition

• Two real numbers or algebraic expressions related by symbols: <, >, ≤, or ≥ form an inequality.
• Examples:
  • Numerical inequalities: 3 < 5; 7 > 5
  • Literal inequalities: x < 5; y > 2; x ≥ 3; y ≤ 4
  • Double inequalities: 3 < x < 5

Examples of Inequalities

1. Strict Inequalities:
   • ax + b < 0
   • ax + b > 0
2. Slack Inequalities:
   • ax + b ≤ 0
   • ax + b ≥ 0
3. Linear Inequalities in Two Variables:
   • ax + by < c
   • ax + by > c

Examples of Problem Solving

• Example 1: Ravi's rice purchase
  • Total amount spent: ₹30x < 200
• Example 2: Reshma's purchase of registers and pens
  • Total amount spent: 40x + 20y ≤ 120

5.3 Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation

Solving Inequalities

• Example: Solve 4x + 3 < 6x + 7
  • Solution: x > -2 (solution set: (-2, ∞))
• Example: Solve 5 - 2x ≤ 8
  • Solution: x ≥ 8 (solution set: [8, ∞))

Graphical Representation

• Fig 5.1: Number line showing solutions of inequalities.
• Fig 5.2: Graphical representation of solutions for x < 3.

5.4 Miscellaneous Examples

• Example 9: Solve -8 ≤ 5x - 3 < 7
  • Solution: -1 ≤ x < 2
• Example 10: Solve the system of inequalities: 3x - 7 < 5 + x < 11 - 5x
  • Solution: 2 ≤ x < 6

Important Diagrams

• Diagram 1: Number line representing x < 3 (open circle at 3).
• Diagram 2: Number line representing x ≥ 1 (closed dot at 1).

Conclusion

• Linear inequalities provide a framework for solving various real-world problems through algebraic expressions and graphical representations.

Common Mistakes and Exam Tips for Linear Inequalities

Common Pitfalls

• Misinterpreting Inequalities: Students often confuse the symbols for inequalities (e.g., using > instead of <). Always double-check the direction of the inequality sign.
• Ignoring the Domain: When solving inequalities, it's crucial to consider the specified domain (natural numbers, integers, real numbers). Solutions can vary significantly based on this.
• Incorrectly Applying Rules: When multiplying or dividing by negative numbers, remember to reverse the inequality sign. This is a common mistake that can lead to incorrect solutions.

Tips for Success

• Graphical Representation: Always represent your solutions on a number line. This visual aid can help you understand the range of solutions better and avoid mistakes.
• Check Your Solutions: After finding the solutions, substitute them back into the original inequality to verify their correctness.
• Practice with Different Domains: Solve inequalities under different conditions (natural numbers, integers, real numbers) to become familiar with how solutions change.
• Use Systematic Approaches: Instead of trial and error, apply systematic techniques for solving inequalities to save time and reduce errors.