Overview
This lecture explains how to solve linear inequalities, emphasizing the steps involved and special rules when working with negative numbers.
Solving Inequalities: Basic Steps
- To solve an inequality, isolate the variable on one side using addition, subtraction, multiplication, or division.
- Treat the inequality sign (<, >, ≤, ≥) like an equals sign, but do not change it unless required by special rules.
Key Rule: Multiplying or Dividing by a Negative
- When you multiply or divide both sides of an inequality by a negative number, you must reverse (flip) the inequality sign.
- Example: If -2x < 8, dividing both sides by -2 gives x > -4.
Step-by-Step Examples
- For x + 3 < 8, subtract 3 from both sides to get x < 5.
- For x / -2 < 5, multiply both sides by -2 (flip the sign) to get x > -10.
- For 7 + 2x ≥ 13: subtract 7 (2x ≥ 6), then divide by 2 to get x ≥ 3.
Alternative Solution Methods
- There may be multiple ways to rearrange and solve an inequality; you can move terms to either side as convenient.
- Both methods will yield the same final solution.
Practice Example Summaries
- For 5 - 3x ≤ -10:
- Method 1: Subtract 5 (−3x ≤ −15), divide by −3 (flip sign) to get x ≥ 5.
- Method 2: Rearrange to 5 ≤ −10 + 3x, solve to get x ≥ 5.
- For 4x + 7 > x - 4:
- Subtract x: 3x + 7 > -4.
- Subtract 7: 3x > -11.
- Divide by 3: x > -11/3.
Key Terms & Definitions
- Inequality — A mathematical statement showing one quantity is greater than, less than, or equal to another.
- Flip the Inequality Sign — Reverse the direction of the sign when multiplying/dividing both sides by a negative number.
Action Items / Next Steps
- Practice solving more inequalities using both one-step and multi-step methods.
- Remember to flip the inequality sign when multiplying or dividing by a negative.