Matematika Wajib Kelas 10 : Persamaan dan Pertidaksamaan Nilai Mutlak VI (pertidaksamaan mutlak)

Introduction

This tutorial focuses on absolute value inequalities, a key concept in mathematics for 10th-grade students. Understanding how to solve these inequalities is crucial for progressing in algebra and preparing for higher-level mathematics. We will explore the three general forms of absolute value inequalities, providing step-by-step instructions and practical tips for solving them.

Step 1: Understanding Absolute Value

• Definition: The absolute value of a number is its distance from zero on the number line, regardless of direction. It is denoted as |x|.
• Key Point: Absolute values are always non-negative. For example, |3| = 3 and |-3| = 3.

Step 2: Recognizing the Forms of Absolute Value Inequalities

There are three general forms of absolute value inequalities:

1. Type 1: |x| < a

   • This means that x lies between -a and a.
   • Solution:
     • Set up the compound inequality: -a < x < a.

2. Type 2: |x| > a

   • This means that x is either less than -a or greater than a.
   • Solution:
     • Use two inequalities: x < -a or x > a.

3. Type 3: |x| ≤ a

   • This indicates that x is within the inclusive range of -a to a.
   • Solution:
     • Set up the compound inequality: -a ≤ x ≤ a.

4. Type 4: |x| ≥ a

   • This indicates that x is either less than or equal to -a or greater than or equal to a.
   • Solution:
     • Use two inequalities: x ≤ -a or x ≥ a.

Step 3: Solving Absolute Value Inequalities

Type 1: |x| < a

• Example: Solve |x| < 3.
• Steps:
  1. Set up the inequalities: -3 < x < 3.
  2. The solution is the interval (-3, 3).

Type 2: |x| > a

• Example: Solve |x| > 2.
• Steps:
  1. Set up the inequalities: x < -2 or x > 2.
  2. The solution is the union of the intervals (-∞, -2) and (2, ∞).

Type 3: |x| ≤ a

• Example: Solve |x| ≤ 4.
• Steps:
  1. Set up the inequalities: -4 ≤ x ≤ 4.
  2. The solution is the interval [-4, 4].

Type 4: |x| ≥ a

• Example: Solve |x| ≥ 1.
• Steps:
  1. Set up the inequalities: x ≤ -1 or x ≥ 1.
  2. The solution is the union of the intervals (-∞, -1] and [1, ∞).

Conclusion

In this tutorial, we covered the fundamental concepts of absolute value inequalities, including their definitions and how to solve each of the four general forms. Remember to set up your inequalities correctly and pay attention to whether the solution should be inclusive or exclusive. Practice with various examples to solidify your understanding. By mastering absolute value inequalities, you will be well-prepared for more complex algebraic concepts in the future.