Below Zero

Numbers live on a line: zero in the middle, positives to the right, negatives to the left. -3 is three steps left of zero, exactly as 3 is three steps right of it. Temperatures, floors below ground and money owed are all negatives, and equations meet them constantly, so their rules are worth stating once, plainly.

Adding a negative walks left: 5 + (-8) = -3. Subtracting is adding the opposite, so subtracting a negative walks right: 2 - (-6) = 2 + 6 = 8. Two minus signs in a row become a plus.

Multiplying or dividing: same signs give a positive, different signs give a negative. (-3) × 4 = -12, (-3) × (-4) = 12, -12 ÷ 4 = -3, -12 ÷ (-4) = 3.

Nothing in the one law changes when a negative appears. You still undo the operation on both sides; the arithmetic just carries a sign. What changes is the answer: from now on x may well turn out negative, and that is a perfectly good solution.