# Question 1032: Piecewise Branch of Absolute Value

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This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth.

R0001-01032 · q-20260928-absolute-value-piecewise-branch-0007



## Original question and givens

For the real number x=6, which branch of the piecewise definition of |x| applies?


- G001 — For the real number x=6, 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.


- [mko-absolute-value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/)





## Step-by-step derivation

S001 · impliesBefore

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 dependenciesRule: absolute_definition · Uses: G001





S002 · impliesBefore

x=6

After

x≥0: Yes

Conditions: none

Determine the sign of the given value rather than guessing from a picture.

Step identifiers and dependenciesRule: sign_test · Uses: G001, S001





S003 · impliesBefore

x=6

After

nonnegative branch x

Conditions: none

Zero satisfies the nonnegative condition; only negative values use −x.

Step identifiers and dependenciesRule: select_absolute_branch · Uses: G001, S002





S004 · calculationBefore

absolute value := mul(1, 6)

After

absolute value = 6

Conditions: none

Multiply by1 or −1 according to the selected branch to obtain a nonnegative value.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003





S005 · impliesBefore

|x|=6

After

result≥0: Yes

Conditions: none

Verify nonnegativity to avoid a negative result from the negative branch.

Step identifiers and dependenciesRule: 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

nonnegative 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: 0f7be61e2c22317a4075561dc822c87d881a50db
- original question SHA-256: 9e59fc98de7ef30d7d6f3011730e7e1cb924d95059b7dae8d3510b45d0723df4
- candidate record SHA-256: daca4f4db85e494b98cdf148066b23f74300eaef74692e2169b62a85bd931b38
- semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496cc
- question-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7ad
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260928-absolute-value-piecewise-branch-0007.json)