The Golden Rule of Inequalities states: Whenever you multiply or divide both sides of an inequality by a negative number, you must flip the inequality symbol.
Understanding the Golden Rule
This rule is crucial when solving inequalities because multiplying or dividing by a negative number changes the direction of the relationship between the two sides of the inequality. Here's a more detailed breakdown:
Why it Matters
- Inequalities use symbols like < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) to compare values.
- Multiplying or dividing by a negative value reverses the order of numbers on the number line. For instance, if 2 < 4, then -2 > -4. This reversal requires us to flip the inequality sign.
The Rule in Action
Here's a table summarizing when to flip the inequality symbol:
| Operation | Condition | Action | Example |
|---|---|---|---|
| Multiplying or Dividing both sides | By a negative number | Flip the inequality sign | If -2x < 6, then x > -3. |
| Multiplying or Dividing both sides | By a positive number | Do not flip the sign | If 2x < 6, then x < 3. |
| Addition or Subtraction | By any number | Do not flip the sign | If x - 2 < 6, then x < 8. |
Examples
-
Example 1:
- Start with: -3x < 12
- Divide both sides by -3 (a negative number):
- Remember to flip the inequality sign: x > -4
-
Example 2:
- Start with: -x ≥ 5
- Multiply both sides by -1 (a negative number):
- Remember to flip the inequality sign: x ≤ -5
Practical Insights
- Always double-check: When you multiply or divide by a negative number, it’s easy to forget to flip the sign. Make it a habit to double-check your work.
- Positive numbers: When dealing with positive numbers, you do not need to flip the inequality sign. You perform the operation and retain the original inequality symbol.
- Addition and Subtraction: Adding or subtracting numbers on both sides of an inequality does not require you to flip the sign.
Summary
The Golden Rule of Inequalities is a fundamental rule: whenever you multiply or divide both sides of an inequality by a negative number, remember to flip the inequality sign. Failing to do so will lead to an incorrect answer.
