Absolute Value Function Basic

Algebra-1

1. Fundamental Concepts

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

2. Key Concepts

Evaluating Absolute Values: $$\text{{|}}3\text{{|}} = 3$$ $$\text{{|}}-5\text{{|}} = 5$$

Solving Equations: If $$\text{{|}}x\text{{|}} = a$$ where $$a > 0$$ , then $$x = a$$ or $$x = -a$$ .

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

3. Examples

Example 1 (Basic)

Problem: Solve $$\text{{|}}x + 4\text{{|}} = 6$$

Step-by-Step Solution:

Set up two equations based on the definition of absolute value: $$x + 4 = 6$$ and $$x + 4 = -6$$
Solve each equation:
- For $$x + 4 = 6$$ , subtract 4 from both sides: $$x = 2$$
- For $$x + 4 = -6$$ , subtract 4 from both sides: $$x = -10$$

Validation: Substitute $$x = 2$$ and $$x = -10$$ into the original equation:

For $$x = 2$$ : $$\text{{|}}2 + 4\text{{|}} = \text{{|}}6\text{{|}} = 6$$ ✓
For $$x = -10$$ : $$\text{{|}}-10 + 4\text{{|}} = \text{{|}}-6\text{{|}} = 6$$ ✓

Example 2 (Intermediate)

Problem: Solve $$\text{{|}}2x - 3\text{{|}} = 7$$

Step-by-Step Solution:

Set up two equations: $$2x - 3 = 7$$ and $$2x - 3 = -7$$
Solve each equation:
- For $$2x - 3 = 7$$ , add 3 to both sides: $$2x = 10$$ , then divide by 2: $$x = 5$$
- For $$2x - 3 = -7$$ , add 3 to both sides: $$2x = -4$$ , then divide by 2: $$x = -2$$

Validation: Substitute $$x = 5$$ and $$x = -2$$ into the original equation:

For $$x = 5$$ : $$\text{{|}}2(5) - 3\text{{|}} = \text{{|}}10 - 3\text{{|}} = \text{{|}}7\text{{|}} = 7$$ ✓
For $$x = -2$$ : $$\text{{|}}2(-2) - 3\text{{|}} = \text{{|}}-4 - 3\text{{|}} = \text{{|}}-7\text{{|}} = 7$$ ✓

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.