The inequality property of division states that when you divide both sides of an inequality by a positive number, the direction of the inequality remains the same. However, if you divide both sides by a negative number, the direction of the inequality must be reversed.
Explanation of the Inequality Property of Division
The division property of inequality is a fundamental rule for manipulating inequalities while preserving their truth. It is essential for solving inequalities in algebra and other mathematical contexts.
Division by a Positive Number
When dividing both sides of an inequality by a positive number, the inequality symbol stays the same.
Example:
If 3x < 9, then dividing both sides by 3 (a positive number) gives:
x < 3
The "<" symbol remains unchanged.
Division by a Negative Number
When dividing both sides of an inequality by a negative number, the inequality symbol must be reversed to maintain the truth of the inequality.
Example:
If -2x < 6, then dividing both sides by -2 (a negative number) gives:
x > -3
Notice that the "<" symbol has changed to ">".
Summary Table
| Operation | Condition | Inequality Symbol | Example |
|---|---|---|---|
| Divide both sides by | Positive Number | Remains the same | 4x > 8 becomes x > 2 |
| Divide both sides by | Negative Number | Reverses | -2x < 4 becomes x > -2 |
Why does the inequality reverse when dividing by a negative number?
Consider the simple inequality:
2 < 4
If we multiply both sides by -1, we get:
-2 > -4
Note that -2 is greater than -4. Multiplying by a negative number changes the sign and, consequently, the relative size of the numbers. The same logic applies to division, as division by a negative number is equivalent to multiplying by its negative reciprocal.
Importance
Understanding and correctly applying this property is crucial for solving inequalities accurately. Failing to reverse the inequality sign when dividing by a negative number will lead to an incorrect solution set.
