Negative Fractions

Solving $\frac{-x}{3} \ge 2$ takes two steps. First multiply both sides by 3. The number 3 is positive, so the sign stays the same, and you get $-x \ge 6$. Then divide both sides by -1. The number -1 is negative, so the sign flips:

$$x \le -6$$

You can also multiply both sides by -3 in one step. The number -3 is negative, so the sign flips once, and you get $x \le -6$ straight away. Either way, you multiply or divide by a negative number once, so the sign flips once.

Count the flips. Each time you multiply or divide by a negative number, the sign flips once. If you did it twice, the sign would flip twice and end up pointing the same way as at the start. Counting the flips is easier than working out the sign at every step.