why did the symbol of the numerator chang from greater to to less than when its divided by -2

When you divide or multiply both sides of an inequality by a **negative number**, the inequality symbol flips direction. This rule arises from the relationship between the order of numbers and how multiplication by negatives affects that order.

Here’s why:

Consider the basic inequality:
\[
3 > 1
\]
If you multiply both sides by \(-2\), you get:
\[
-2 \times 3 = -6 \quad \text{and} \quad -2 \times 1 = -2
\]
But now, \(-6\) is not greater than \(-2\). Instead, \(-6\) is less than \(-2\), so the inequality must flip:
\[
-6 < -2
\]

Thus, dividing or multiplying by a negative number reverses the inequality symbol to maintain the truth of the inequality.

Would you like more details on this concept or examples to further clarify? 

Here are 5 related questions to expand your understanding:
1. What happens when you divide an inequality by a positive number?
2. How do you solve inequalities involving absolute values?
3. Why does multiplying by a negative number reverse inequalities but not equalities?
4. Can you explain the difference between solving linear inequalities and quadratic inequalities?
5. What are common mistakes when solving inequalities?

**Tip:** Always remember to flip the inequality sign when multiplying or dividing by a negative number, especially in multi-step problems!

Understand why the inequality symbol flips when dividing or multiplying by a negative number. Learn the properties of inequalities and see examples of how this rule works. Suitable for students in Grades 6-8.

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