Yes, you can divide inequalities, but there's a crucial rule to remember regarding negative numbers.
Dividing inequalities is similar to dividing equations, with one important exception. The rule that governs this process is summarized as:
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Dividing by a positive number: If you divide both sides of an inequality by a positive number, the inequality sign remains the same.
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Dividing by a negative number: If you divide both sides of an inequality by a negative number, you must reverse the inequality sign. For example,
>becomes<, and<becomes>. (Rule 3 from provided reference.)
Examples:
Dividing by a Positive Number:
Suppose we have the inequality:
4x > 8
Dividing both sides by 4 (a positive number) gives us:
x > 2
The inequality sign remains the same.
Dividing by a Negative Number:
Suppose we have the inequality:
-2x < 6
Dividing both sides by -2 (a negative number) gives us:
x > -3
Notice that the inequality sign flipped from < to >. This sign change is essential for maintaining the truth of the inequality.
Summary Table:
| Operation | Condition | Inequality Sign | Example |
|---|---|---|---|
| Divide by a positive number | c > 0 |
No change | 2x < 4 => x < 2 |
| Divide by a negative number | c < 0 |
Reverse | -3x > 9 => x < -3 |
Important Considerations:
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Dividing by zero: Division by zero is undefined, so you can never divide an inequality (or any equation) by zero.
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Variable sign: When you don't know the sign of the variable, the sign change cannot be determined, and the inequality cannot be determined.
