Absolute Value as Distance

Absolute value strips away direction and keeps size. On the number line, -7 sits 7 steps left of zero and 7 sits 7 steps right of zero, so both get absolute value 7: distance never cares which way you walked, only how far. That's why |-5| = 5 — a negative number is just a positive distance pointed the other way, and the bars erase the pointing.

To find the distance between two points a and b, subtract them and take the absolute value: |a - b|. The subtraction finds the gap between the two locations, and the bars make sure that gap comes out positive even if you subtract in the 'wrong' order. Try swapping the points: |a - b| and |b - a| always give the same number, because b - a is just the negative of a - b, and absolute value erases that sign. This is why distance is symmetric — the distance from your house to school equals the distance from school to your house.

This idea connects directly to the rest of the number-and-quantity items on the test, where absolute value shows up again in inequalities and in the modulus of complex numbers, which is really just distance from the origin on the complex plane. Get comfortable with |a - b| as 'distance between a and b' now, and that complex-number question becomes the same skill in a new costume.