Inequalities 1 | CAT Preparation 2024 | Algebra | Quantitative Aptitude

Introduction

This tutorial is designed for students preparing for the CAT 2024 exam, focusing on understanding and solving inequalities in algebra. By following these steps, you will enhance your quantitative aptitude skills, crucial for performing well in the exam.

Step 1: Understanding Inequalities

• Definition: Inequalities express a relationship where one value is greater than or less than another.
• Types of Inequalities:
  • Linear Inequalities: Involves linear expressions. Example: x + 5 > 10.
  • Quadratic Inequalities: Involves quadratic expressions. Example: x^2 - 4 < 0.

Practical Tips

• Familiarize yourself with inequality symbols:
  • Greater than: >
  • Less than: <
  • Greater than or equal to: ≥
  • Less than or equal to: ≤

Step 2: Solving Linear Inequalities

1. Isolate the variable:

   • Aim to get the variable on one side of the inequality.
   • Example: From x + 5 > 10, subtract 5 from both sides to find x > 5.

2. Graphing the solution:

   • Use a number line to represent the solution.
   • For x > 5, place an open circle on 5 and shade to the right.

Common Pitfalls

• Remember that multiplying or dividing both sides by a negative number reverses the inequality sign.

Step 3: Solving Quadratic Inequalities

1. Factor the quadratic expression:

   • Example: For x^2 - 4 < 0, factor to (x - 2)(x + 2) < 0.

2. Determine the critical points:

   • Set each factor to zero: x - 2 = 0 or x + 2 = 0, leading to x = -2 and x = 2.

3. Test intervals:

   • Choose test points in the intervals determined by the critical points (-∞, -2), (-2, 2), and (2, ∞).
   • Check the sign of the product in each interval.

Practical Tips

• Use a table to organize test points and results for clarity.

Step 4: Application of Inequalities

• Word Problems: Inequalities often appear in real-world scenarios such as budgeting, planning, and resource allocation.
• Practice Problems: Regularly solve a variety of problems to strengthen your understanding and application skills.

Conclusion

Understanding and solving inequalities is essential for the CAT exam. Focus on mastering both linear and quadratic inequalities by practicing regularly and applying these concepts in various scenarios. As you prepare, consider reviewing related topics in quantitative aptitude to build a solid foundation for success in the exam.