1. The problem asks: When does an equality sign flip? 2. Actually, equality signs (=) do not flip in mathematics. They always indicate that two expressions are exactly equal. 3. However, inequality signs (>, <, \geq, \leq) can flip under certain conditions. 4. The key rule is: When you multiply or divide both sides of an inequality by a negative number, the inequality sign flips direction. 5. For example, if you have $a < b$ and multiply both sides by $-1$, it becomes $-a > -b$. 6. This rule does not apply to equality signs; equalities remain equal regardless of multiplication or division by negative numbers. 7. So, the equality sign never flips; only inequality signs flip when multiplied or divided by negative numbers.