S001 · implies
Before
x∈R
After
|x|=x if x≥0; |x|=−x if x<0
Conditions: none
The sign branches cover every real number, with0 in the nonnegative branch.
Step identifiers and dependencies
Rule: absolute_definition · Uses: G001
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Question 1039: Piecewise Branch of Absolute Value
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R0001-01039 · q-20260928-absolute-value-piecewise-branch-0014
Original question and givens
For the real number x=-1, which branch of the piecewise definition of |x| applies?
G001— For the real number x=-1, which branch of the piecewise definition of |x| applies?
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
Step-by-step derivation
S002 · implies
Before
x=-1
After
x≥0: No
Conditions: none
Determine the sign of the given value rather than guessing from a picture.
Step identifiers and dependencies
Rule: sign_test · Uses: G001, S001
S003 · implies
Before
x=-1
After
negative branch -x
Conditions: none
Zero satisfies the nonnegative condition; only negative values use −x.
Step identifiers and dependencies
Rule: select_absolute_branch · Uses: G001, S002
S004 · calculation
Before
absolute value := mul(-1, -1)
After
absolute value = 1
Conditions: none
Multiply by1 or −1 according to the selected branch to obtain a nonnegative value.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
|x|=1
After
result≥0: Yes
Conditions: none
Verify nonnegativity to avoid a negative result from the negative branch.
Step identifiers and dependencies
Rule: absolute_result_guard · Uses: G001, S004
Reverse checks
S005· premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S004· premise_and_result_audit — Recompute absolute value with exact rational arithmetic, not rounded decimals.S003· premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S002· premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.S001· premise_and_result_audit — Use ≥0 rather than >0 so that0 is not omitted.
Result
negative branch -x
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.
Source hashes and review receipts
input commit: 0f7be61e2c22317a4075561dc822c87d881a50dboriginal question SHA-256: 88c46e689740ba253b0e341f1936416b51f5892946fbc46d85cd3658722117a8candidate record SHA-256: e31e4b052680ba1ec70950dfcfb134fb0dfeee709abd3658cb6efbed33973e37semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED