Comprehensive Guide to Absolute Value

This page provides a thorough overview of the concept of absolute value, its properties, and various problem-solving techniques. It's an essential resource for students preparing for exams or seeking to deepen their understanding of this fundamental mathematical concept.

Definition: The absolute value of a number is its distance from zero on the number line, regardless of whether the number is positive or negative.

Key Properties of Absolute Value

1. Non-negativity: The absolute value is always non-negative.

   Example: |-5| = 5, |3| = 3

2. Equality: |x| = a implies x = a or x = -a

   Example: If |x| = 3, then x could be 3 or -3

3. Inequalities:

   • |x| < a means x is between -a and a, i.e., x ∈ (-a, a)
   • |x| > a means x is less than -a or greater than a, i.e., x ∈ (-∞, -a) ∪ (a, +∞)

4. Distance representation: |x - a| = t represents points at distance t from a on the number line

5. Symmetry: |x + a| = |x - (-a)| shows symmetry around zero

Solving Equations and Inequalities

The page demonstrates various techniques for solving equations and inequalities involving absolute values:

Example: |x + 3| = 2 Solution: x + 3 = 2 or x + 3 = -2 x = -1 or x = -5

Example: |x - 4| ≤ 6 Solution: -6 ≤ x - 4 ≤ 6 -2 ≤ x ≤ 10

Advanced Properties and Techniques

The guide also covers more advanced topics:

• Combining absolute values: |x·y| = |x|·|y|
• Squaring absolute values: |x|² = x²
• Dealing with nested absolute values

Highlight: Pay special attention to the domain of x when solving absolute value problems, especially when dealing with square roots or fractions.

This comprehensive overview of własności wartości bezwzględnej (properties of absolute value) and wartość bezwzględna zadania (absolute value problems) provides students with the tools they need to tackle complex problems and excel in their mathematical studies.