Do you only flip the inequality sign when dividing a negative number?

Question

Does it happen in multiplication too?

Answer 1

As I mentioned in the comments, multiplication and division are essentially the same thing. You can do some experimenting as see what "should" happen. For example:

You know that $2 < 3$. If I multiply both sides by $-2$ which direction should the inequality go? $-4 < -6$ or $-4 > -6$? This would suggest that you should flip the inequality if you multiply by a negative number.

If you want to know why, think about the numbers on a number line. You say that $2 < 3$ because $2$ is to the left of $3$ on the number line. When you multiply (or divide) by a negative value, the numbers are reflected over the zero on the number line (and possibly scaled). For example, multiplying both numbers by $-1$ gives you $-2$ and $-3$. The result is that the number that was closer to $0$ (the $2$) is still closer to $0$... but it was on the left and is now on the right (because of the reflection). So the number that was bigger is now smaller--so the inequality flips.

Answer 2

When you have an inequality, for instance $a>b$, and you multiply or divide by a negative number, the inequality "flips" to $-ak<-bk$, where $k$ is the number you multiply by (which is just dividing by $1/k$. One way to think about this is that the greater than sign is measuring "rightness" of numbers - that is, how far right they are on the number line. When you multiply by a positive number you're just stretching or squishing the number line, so which one is farther to the right doesn't change, but when you stretch or squish and flip it upside down (multiplying or dividing by a negative number) the one that used to be on the right is now farther to the left, so the inequality sign flips.