Absolute Value Function Basic

Algebra-1

1. Fundamental Concepts

Definition: The absolute value function, denoted as , represents the non-negative value of without regard to sign.
Graphical Representation: The graph of is a V-shaped curve with its vertex at the origin (0,0).
Properties: For any real number , and .

2. Key Concepts

Evaluating Absolute Values:

Solving Equations: If where , then or .

Application: Used in various fields such as physics for distance calculations and in engineering for signal processing.

3. Examples

Example 1 (Basic)

Problem: Solve

Step-by-Step Solution:

Set up two equations based on the definition of absolute value: and
Solve each equation:
- For , subtract 4 from both sides:
- For , subtract 4 from both sides:

Validation: Substitute and into the original equation:

For : ✓
For : ✓

Example 2 (Intermediate)

Problem: Solve

Step-by-Step Solution:

Set up two equations: and
Solve each equation:
- For , add 3 to both sides: , then divide by 2:
- For , add 3 to both sides: , then divide by 2:

Validation: Substitute and into the original equation:

For : ✓
For : ✓

4. Problem-Solving Techniques

Isolate the Absolute Value: Always start by isolating the absolute value expression on one side of the equation.
Consider Both Cases: After isolating, set up two separate equations—one assuming the expression inside the absolute value is positive, and the other assuming it is negative.
Check Solutions: Always verify solutions by substituting them back into the original equation to ensure they satisfy the conditions.