Exercise 3: Predicting ~(-5) and Explaining Double Complement — Possible Solution
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STEP 1: PREDICT ~(-5) USING ~x = -x-1
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~x = -x - 1, with x = -5:
~(-5) = -(-5) - 1 = 5 - 1 = 4

Predicted value: 4

STEP 2: VERIFY
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Computing ~(-5) directly gives 4. The prediction matches exactly.

STEP 3: WHY ~~x ALWAYS RETURNS x
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Applying this chapter's own identity twice: ~x = -x-1, and then
~(~x) = ~(-x-1) = -(-x-1)-1 = (x+1)-1 = x.

The second negation and the second "-1" exactly cancel the first
pair's own effect - applying the same "negate and subtract 1"
transformation twice reverses itself completely, in the same way
that applying "multiply by -1" twice returns the original number:
each transformation exactly undoes what the other one does,
algebraically, for any x.

RESULT
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~(-5) = 4, confirmed. And ~~x = x always, because the ~x=-x-1
identity, applied twice in sequence, algebraically cancels itself
out completely regardless of what x actually is.

WHY THIS WORKS AS AN ANSWER
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The prediction is made purely from this chapter's own stated
identity (not by simply computing the answer first and working
backward), and the double-complement explanation is derived
algebraically from that same identity applied twice, rather than
just asserting "complementing twice cancels out" as an unexplained
fact.