The Line and the Balance

Unit 3 runs on one law — whatever is done to one side is done to the other — and everything else is bookkeeping about direction, distance, and the pictures solutions make.

The balance law carries the whole unit. It solves 3x + 7 = 19 in two inverse moves and lands on x = 4, and the substitution check — 12 + 7 = 19 — is the receipt, because a solution is verified by checking, never by trusting the steps. The same law sorts equations into three kinds by their solution counts: all values, some, none. Rational equations pay an extra toll: the LCD clears denominators, but clearing can smuggle in values the original forbids — x/(x − 2) = 2/(x − 2) offers x = 2, the one number both denominators reject — so the extraneous audit runs at the original, always. Then the unit draws its pictures. Two crossed axes turn pairs of numbers into points; slope compresses a line's whole direction into one ratio, 6 over 3 through (1, 2) and (4, 8); and one line wears many costumes — slope-intercept, point-slope, the two-point build — each unwrapping to the others. Parallel lines share the ratio; perpendicular ones multiply theirs to −1. Inequalities inherit the balance law with a single statute added: a negative multiplier flips the symbol, and answers arrive as intervals, whole sets in one stroke. Absolute value closes the unit by turning distance into algebra — within 3 of 7 is |x − 7| ≤ 3 — with less-than trapping an interval and greater-than banishing to rays. Both of the unit's error twins die by their own test values: the kept sign endorses x = −4 and gets 8 < 6, false; the swapped cases welcome x = 6 into |x| < 5, false. The check is the conscience of the unit, and the errors never survive it.