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 1091: Solution Classification of an Absolute-Value Equation
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R0001-01091 · q-20260928-absolute-value-equation-class-0016
Original question and givens
Over the real numbers, classify the solution set of |x|=11.
G001— Over the real numbers, classify the solution set of |x|=11.
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|=11
After
|x|≥0 for every real x
Conditions: none
From the piecewise definition, x or −x is nonnegative on its respective branch.
Step identifiers and dependencies
Rule: absolute_nonnegativity · Uses: G001, S001
S003 · calculation
Before
negative candidate := mul(-1, 11)
After
negative candidate = -11
Conditions: none
In the negative branch, −x=c gives x=−c.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S002
S004 · implies
Before
x≥0 gives x=11; x<0 would give x=-11
After
distinct permitted roots=[-11, 11]
Conditions: none
The sign branches are exhaustive; for c=0 the formal candidates coincide and the negative branch adds no negative root.
Step identifiers and dependencies
Rule: exhaustive_sign_cases · Uses: G001, S003
S005 · implies
Before
[-11, 11]
After
each displayed root has |x|=11
Conditions: none
Substitute every candidate and check its branch sign to obtain every and only solution.
Step identifiers and dependencies
Rule: substitution_and_exhaustion · 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence.S003· premise_and_result_audit — Recompute negative candidate with exact rational arithmetic, not rounded decimals.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
two solutions x=-11, 11
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: c483c2b4e55268f24ead8f4a602d23430e3572a0f3f0d471aca5946bfc32ac27candidate record SHA-256: a831cb03aff83d5dc60c299c1151d9dec3536e2d6524c0d1cc3a10901d251961semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED