Solve Absolute Value Inequalities

Algebra-2

1. Fundamental Concepts

Definition: Absolute value inequalities involve expressions where the absolute value of a variable is compared to another number using inequality symbols.
Types: There are two main types of absolute value inequalities: "less than" (|x| < a) and "greater than" (|x| > a).
Solution Sets: The solutions to these inequalities can be expressed as intervals on the number line.

2. Key Concepts

Basic Rule: $$\text{{If}} \ |x| < a, \text{{then}} -a < x < a$$

Degree Preservation: $$\text{{If}} \ |x| > a, \text{{then}} x < -a \text{{ or }} x > a$$

Application: Used to solve real-world problems involving constraints and boundaries

3. Examples

Example 1 (Basic)

Problem: Solve $$|2x + 3| < 5$$

Step-by-Step Solution:

Set up the compound inequality: $$-5 < 2x + 3 < 5$$
Solve for \(x\):
- Subtract 3 from all parts: $$-8 < 2x < 2$$
- Divide by 2: $$-4 < x < 1$$

Validation: Substitute \(x = 0\) → Original: \(|2(0) + 3| = 3 < 5\); Simplified: \(0 < 0 < 1\) ✓

Example 2 (Intermediate)

Problem: Solve $$|3x - 4| > 2$$

Step-by-Step Solution:

Set up the compound inequality:
- \(3x - 4 < -2\) or \(3x - 4 > 2\)
Solve each part:
- For \(3x - 4 < -2\): Add 4 to both sides: \(3x < 2\), then divide by 3: \(x < \frac{2}{3}\)
- For \(3x - 4 > 2\): Add 4 to both sides: \(3x > 6\), then divide by 3: \(x > 2\)

Validation: Substitute \(x = 0\) → Original: \(|3(0) - 4| = 4 > 2\); Simplified: \(0 < \frac{2}{3}\) ✓

4. Problem-Solving Techniques

Visual Strategy: Graph the inequality on a number line to visualize the solution set.
Error-Proofing: Always check the endpoints of the interval(s) in the original inequality.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of different types of absolute value inequalities.