Matematika Dasar II: Pertidaksamaan Nilai Mutlak

Introduction

This tutorial covers the fundamental concepts of absolute value inequalities, building upon previous lessons in basic mathematics. Understanding these inequalities is essential for solving various mathematical problems and can be applied in real-life situations like budgeting, measurements, and more.

Step 1: Understanding Absolute Value

• Definition: The absolute value of a number represents its distance from zero on a number line, regardless of direction.
• Notation: The absolute value of a number ( x ) is denoted as ( |x| ).
  • Examples:
    • ( |5| = 5 )
    • ( |-3| = 3 )

Step 2: Formulating Absolute Value Inequalities

• Inequality Basics: An absolute value inequality can be expressed as ( |x| < a ) or ( |x| > a ).
• Interpretation:
  • For ( |x| < a ):
    • This means ( -a < x < a ).
  • For ( |x| > a ):
    • This translates to ( x < -a ) or ( x > a ).

Step 3: Solving Absolute Value Inequalities

• Step-by-Step Approach:
  1. Identify the inequality: Determine if it's a less than or greater than inequality.
  2. Split into cases:
     • For ( |x| < a ):
       • Solve the two inequalities: ( -a < x < a ).
     • For ( |x| > a ):
       • Solve: ( x < -a ) or ( x > a ).
  3. Graph the solution on a number line for visual representation.

Step 4: Practical Examples

• Example 1: Solve ( |x| < 4 )
  • Solution:
    • ( -4 < x < 4 )
• Example 2: Solve ( |x| > 2 )
  • Solution:
    • ( x < -2 ) or ( x > 2 )

Step 5: Common Pitfalls to Avoid

• Misinterpretation of Symbols: Ensure you correctly identify whether the inequality is a less than or greater than.
• Neglecting to split cases: Always remember to split the inequality into its respective cases for accurate solutions.

Conclusion

In this tutorial, we covered the definition of absolute value, how to formulate and solve absolute value inequalities, and provided practical examples to illustrate these concepts. Understanding these principles will enhance your mathematical problem-solving skills and prepare you for more advanced topics. Next, consider practicing with additional inequalities to reinforce your understanding.