Absolute Value Inequalities Vs. Equations

Algebra-1

1. Fundamental Concepts

Definition: Absolute value inequalities involve expressions where the absolute value of a variable is compared to another number using inequality symbols.
Absolute Value: The absolute value of a number is its distance from zero on the number line, always non-negative.
Inequality Symbols: Commonly used symbols include <, >, ≤, and ≥.

2. Key Concepts

Basic Rule: $$|x| \lt a \implies -a \lt x \lt a$$

Degree Preservation: The solution set for $$|x| \lt a$$ is an interval centered at 0 with length 2a.

Application: Used in various real-world scenarios such as error margins in measurements.

3. Examples

Example 1 (Basic)

Problem: Solve $$|2x + 3| \lt 5$$

Step-by-Step Solution:

Isolate the absolute value expression: $$|2x + 3| \lt 5$$
Set up the compound inequality: $$-5 \lt 2x + 3 \lt 5$$
Solve for x:
- Subtract 3 from all parts: $$-8 \lt 2x \lt 2$$
- Divide by 2: $$-4 \lt x \lt 1$$

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

Example 2 (Intermediate)

Problem: Solve $$|3x - 4| \geq 7$$

Step-by-Step Solution:

Isolate the absolute value expression: $$|3x - 4| \geq 7$$
Set up two separate inequalities:
- $$3x - 4 \geq 7$$
- $$3x - 4 \leq -7$$
Solve each inequality:
- For $$3x - 4 \geq 7$$ : Add 4 to both sides: $$3x \geq 11$$ ; Divide by 3: $$x \geq \frac{11}{3}$$
- For $$3x - 4 \leq -7$$ : Add 4 to both sides: $$3x \leq -3$$ ; Divide by 3: $$x \leq -1$$

Validation: Substitute x = 4 → Original: |3(4) - 4| = 8; Simplified: 8 ≥ 7 ✓

4. Problem-Solving Techniques

Visual Strategy: Use a number line to visualize the solution set.
Error-Proofing: Always check the validity of solutions by substituting back into the original inequality.
Concept Reinforcement: Practice with a variety of problems to reinforce understanding of different types of absolute value inequalities.