The Flip

Here is the one move that behaves differently. Start from something true: 2 < 5. Multiply both sides by -1: -2 and -5. Which is bigger now? On the number line -2 sits to the right of -5, so -2 > -5. The order reversed.

2 is left of 5, but -2 is right of -5: multiplying by a negative turns the order around

So multiplying or dividing both sides of an inequality by a negative number flips the sign: < becomes >, ≤ becomes ≥, and the other way round. -2x < 6: divide both sides by -2, and flip.

x > -3

Check: 0 is above -3, and -2 × 0 = 0 is less than 6. Good. Had the sign not flipped, the answer x < -3 would say -10 works, but -2 × -10 = 20 is not less than 6. The flip is not a convention; it is what the numbers do.