KULIAH KALKULUS Chapter 0: Pertidaksamaan dan Nilai Mutlak (Part 2)

Introduction

This tutorial is designed to provide a clear and concise guide on understanding inequalities and absolute values, which are foundational concepts in calculus. Drawing from the video series by Zero Tutorial Matematika, this guide will help you grasp these concepts step-by-step, making them easier to apply in mathematical problems.

Step 1: Understanding Inequalities

Inequalities express the relationship between two values when they are not equal. They can be categorized into several types:

• Linear Inequalities: An inequality of the form ax + b < c or ax + b > c.
• Rational Inequalities: Involves fractions and can be complex, requiring careful consideration of the numerator and denominator.

Practical Tips

• Always pay attention to the direction of the inequality sign when manipulating inequalities.
• When multiplying or dividing by a negative number, reverse the inequality sign.

Step 2: Solving Linear Inequalities

To solve a linear inequality, follow these steps:

1. Isolate the variable:

   • Use addition or subtraction to move constant terms to one side.
   • Use multiplication or division to isolate the variable.

2. Graph the solution:

   • Plot the solution on a number line.
   • Use open circles for < or > and closed circles for ≤ or ≥.

Common Pitfalls

• Forgetting to change the inequality sign when multiplying or dividing by a negative number.
• Misinterpreting the solution set on a number line.

Step 3: Understanding Absolute Values

The absolute value of a number represents its distance from zero on the number line, regardless of direction. It is denoted as |x|.

Key Points

• |x| = x if x ≥ 0
• |x| = -x if x < 0

Step 4: Solving Absolute Value Equations

To solve equations involving absolute values:

1. Set up two cases:

   • Case 1: Solve the equation as is (remove the absolute value).
   • Case 2: Solve the equation with a negative sign (set the expression inside the absolute value equal to the negative of the other side).

2. Solve each case:

   • Find the solutions for both cases.

3. Check your solutions:

   • Substitute back into the original equation to verify.

Example

For the equation |x - 3| = 5:

• Case 1: x - 3 = 5 → x = 8
• Case 2: x - 3 = -5 → x = -2

Conclusion

Understanding inequalities and absolute values is crucial for mastering calculus. By following the steps outlined in this guide, you can effectively solve and graph inequalities and absolute value equations. To further your learning, consider practicing with various examples and challenging problems. Always reach out for help if you encounter difficulties, and continue exploring advanced topics in calculus.