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 1079: Solution Classification of an Absolute-Value Equation
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R0001-01079 · q-20260928-absolute-value-equation-class-0004
Original question and givens
Over the real numbers, classify the solution set of |x|=-1.
G001— Over the real numbers, classify the solution set of |x|=-1.
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 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 · implies
Before
c=-1<0
After
nonnegative left side cannot equal this right side
Conditions: none
A nonnegative number cannot equal a negative one, so no real solution exists; sampling guesses is unnecessary.
Step identifiers and dependencies
Rule: negative_rhs_obstruction · Uses: G001, S002
S004 · implies
Before
contradiction for every x∈R
After
no solution
Conditions: none
By the definition of solution set, this contradiction gives the empty set.
Step identifiers and dependencies
Rule: empty_solution_set · Uses: G001, S003
Reverse checks
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 — 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
no solution
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: 03d4ac85666e6fdfe7358a4382b03afcf0c4c037267c11e5ba5ac7fff4a3e188candidate record SHA-256: e38d8a8563d1084ae0cac4f34018a2695d065ced54a4b0f2af14af9712e73d5bsemantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED