Solve Multi-Step Inequalities

Algebra-1

1. Fundamental Concepts

Definition: Multi-step inequalities involve solving inequalities that require more than one step to isolate the variable.
Operations: Include addition, subtraction, multiplication, and division of both sides by positive or negative numbers.
Sign Reversal: Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

2. Key Concepts

Basic Rule: $$3x + 4 \cdot 2 > 10$$

Distributive Property: $$2(x - 3) + 4 < 6$$

Application: Used in real-world scenarios such as budgeting, physics, and engineering problems.

3. Examples

Example 1 (Basic)

Problem: Solve $$3x + 4 > 10$$

Step-by-Step Solution:

Subtract 4 from both sides: $$3x > 6$$
Divide both sides by 3: $$x > 2$$

Validation: Substitute x=3 → Original: 3(3) + 4 = 13 > 10 ✓

Example 2 (Intermediate)

Problem: Solve $$2(x - 3) + 4 < 6$$

Step-by-Step Solution:

Distribute the 2: $$2x - 6 + 4 < 6$$
Simplify: $$2x - 2 < 6$$
Add 2 to both sides: $$2x < 8$$
Divide both sides by 2: $$x < 4$$

Validation: Substitute x=3 → Original: 2(3 - 3) + 4 = 4 < 6 ✓

4. Problem-Solving Techniques

Isolation Strategy: Always start by isolating the variable on one side of the inequality.
Sign Awareness: Be cautious when multiplying or dividing by negative numbers, as it changes the direction of the inequality.
Verification Step: After solving, substitute a value from the solution set back into the original inequality to verify correctness.