Solving Absolute Value Inequalities

This page deals with techniques for solving nierówności z wartością bezwzględną (inequalities with absolute value).

Example: Solve |x + 1| < 2

The solving process for inequalities is similar to equations but results in intervals rather than specific values:

1. For |x| < a: -a < x < a
2. For |x| > a: x < -a or x > a

Highlight: The solution to an absolute value inequality is often represented as an interval or a union of intervals on the real number line.

The page includes various types of inequalities, including those with "less than or equal to" and "greater than or equal to" signs.

Applications of Absolute Value

This page explores real-world applications of absolute value concepts, demonstrating their practical relevance.

Example: Using absolute value to calculate the distance between two points on a number line.

The applications cover various fields, including physics, economics, and engineering, showing how absolute value is used to model real situations.

Highlight: Understanding absolute value is crucial for solving many practical problems, especially those involving distances or deviations from a central value.

Introduction to Absolute Value

The concept of absolute value is introduced, defining it for real numbers.

Definition: The absolute value of a real number x is:

• x if x is non-negative
• -x if x is negative

Key properties of absolute value are presented:

1. |x| ≥ 0 for any real number x
2. |x| = |-x| for any real number x

Example: |-2| = 2 and |3| = 3

The page includes various calculations involving absolute values, demonstrating how to simplify expressions containing absolute value symbols.

Highlight: Understanding the basic properties of absolute value is crucial for solving more complex problems involving równania z wartością bezwzględną and nierówności z wartością bezwzględną.

Solving Absolute Value Equations

This section focuses on techniques for solving równania z wartością bezwzględną (equations with absolute value).

Example: Solve |x - 2| = 3

The general approach for solving such equations is presented:

1. Consider two cases: x - 2 ≥ 0 and x - 2 < 0
2. Solve each case separately
3. Combine the solutions

Highlight: The solution to an absolute value equation often consists of two values equidistant from the number inside the absolute value symbols.

The page also covers more complex equations involving multiple absolute value terms or additional algebraic operations.

Graphical Representation of Absolute Value

This section introduces the graphical interpretation of absolute value functions and their equations or inequalities.

Definition: The graph of y = |x| is V-shaped, with the vertex at the origin.

The page demonstrates how to sketch graphs of more complex absolute value functions and how to interpret solutions to equations and inequalities graphically.

Example: The graph of y = |x - 2| is the same V-shape as y = |x|, but shifted 2 units to the right.

Highlight: Graphing absolute value functions can provide visual insights into the solutions of równania z wartością bezwzględną and nierówności z wartością bezwzględną.