A worked example: |-7.5| and |-3.2|
Feed -7.5 into the absolute value formula and the sign simply drops off: |-7.5| = 7.5. Feed in -3.2 and the same thing happens: |-3.2| = 3.2. In both cases the number's distance from zero on the number line is what survives — -7.5 sits 7.5 units to the left of zero, and 7.5 units is exactly what the absolute value reports back, with no minus sign attached.
The formula: keep positives, flip negatives
The rule is a simple branch — |x| = x if x is zero or positive, and |x| = -x (which flips a negative number positive) if x is negative. Run 42 through it and nothing changes, because 42 is already non-negative: |42| = 42. Run -12 through it and the sign flips: |-12| = 12. Either way, the output can never be negative — that's the entire point of the operation.
A second example: comparing -12 and 42
Line up |-12| = 12 and |42| = 42 side by side. Even though -12 and 42 are far apart on the number line and one is negative while the other is positive, absolute value only cares about magnitude, not direction — it can't be used to tell which original number was larger or whether it was negative to begin with. That information is thrown away on purpose; what remains is a pure distance-from-zero measurement.
Why throwing away the sign is useful
Plenty of real calculations only care about how far off something is, not which direction it's off in — a percent error calculation doesn't want a negative result just because the actual value happened to be below the target, and neither does a check for how far a measurement drifted from spec. Absolute value is the tool that converts "off by -7.5" and "off by 7.5" into the same answer: off by 7.5. The companion article on why a vector's magnitude reduces to absolute value in one dimension shows this same distance-only idea showing up in a very different-looking formula.