The Flip

Multiplying by a negative number works differently. Start from a true statement: $2 < 5$. Multiply both sides by -1. The left side becomes $-2$ and the right side becomes $-5$. Now compare them. On the number line, -2 is to the right of -5, so $-2 > -5$. The order reversed.

2 is to the left of 5, but -2 is to the right of -5. Multiplying by a negative number reverses the order.

When you multiply or divide both sides of an inequality by a negative number, the sign flips. A < becomes a >, a ≤ becomes a ≥, and the other way round. To solve $-2x < 6$, divide both sides by -2 and flip the sign:

$$x > -3$$

Check the answer with $x = 0$. It satisfies $x > -3$, and in the original inequality $-2\cdot 0 = 0$, which is less than 6. If you did not flip the sign, you would get $x < -3$, which includes -10. But $-2\cdot \left(-10\right) = 20$, and 20 is not less than 6, so that answer is wrong.