I remember to do it about half the time and I don't know the reason, which is probably why I forget.
Accepted answer
Because multiplying or dividing by a negative **reverses order on the number line**.
Start with something obviously true: `−2 < 3`. Multiply both sides by `−1`: you get `2` and `−3`. Is `2 < −3`? No — `2 > −3`. The relationship flipped, because negating reflects every number across zero, and reflection reverses which one is further right.
So when you divide `−2x < 6` by `−2`, you get `x > −3`, not `x < −3`.
Two things that trip people up:
- **Adding or subtracting a negative doesn't flip anything.** Only multiplying or dividing does. `x + 5 < 3` → `x < −2`, no flip.
- It's about the sign of *what you divide by*, not the signs in the problem. `2x < −6` divides by positive 2 → `x < −3`, no flip.
When in doubt, test one number from your answer in the original inequality.