Absolute Value Equations - Algebra-1

1. Fundamental Concepts

Definition: An absolute value equation is an equation involving the absolute value of a variable, denoted as \(\lvert x \rvert\).
Absolute Value: The absolute value of a number \(x\) is its distance from zero on the number line, \(\lvert x \rvert = x\) if \(x \geq 0\) and \(\lvert x \rvert = -x\) if \(x < 0\).
Solutions: Absolute value equations can have zero, one, or two solutions depending on the structure of the equation.

2. Key Concepts

Basic Rule: \(\lvert x \rvert = a\) implies \(x = a\) or \(x = -a\)

No Solution: \(\lvert x \rvert = -a\) has no solution if \(a > 0\)

Application: Used in real-world scenarios such as distance, magnitude, and error analysis

3. Examples

Example 1 (Basic)

Problem: Solve \(\lvert x \rvert = 5\)

Step-by-Step Solution:

Apply the basic rule: \(x = 5\) or \(x = -5\)

Validation: Substitute \(x = 5\) → \(\lvert 5 \rvert = 5\); Substitute \(x = -5\) → \(\lvert -5 \rvert = 5\) ✓

Example 2 (Intermediate)

Problem: Solve \(\lvert 2x - 3 \rvert = 7\)

Step-by-Step Solution:

Set up two equations: \(2x - 3 = 7\) and \(2x - 3 = -7\)
Solve each equation:
- \(2x - 3 = 7\)
  1. Add 3 to both sides: \(2x = 10\)
  2. Divide by 2: \(x = 5\)
- \(2x - 3 = -7\)
  1. Add 3 to both sides: \(2x = -4\)
  2. Divide by 2: \(x = -2\)

Validation: Substitute \(x = 5\) → \(\lvert 2(5) - 3 \rvert = \lvert 10 - 3 \rvert = \lvert 7 \rvert = 7\); Substitute \(x = -2\) → \(\lvert 2(-2) - 3 \rvert = \lvert -4 - 3 \rvert = \lvert -7 \rvert = 7\) ✓

4. Problem-Solving Techniques

Isolate the Absolute Value: Always isolate the absolute value expression on one side of the equation before solving.
Consider Both Cases: Set up and solve both the positive and negative cases for the absolute value expression.
Check Solutions: Always verify the solutions by substituting them back into the original equation to ensure they are valid.