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 1051: Distance on the Real Number Line
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R0001-01051 · q-20260928-absolute-value-distance-0001
Original question and givens
On the real number line, a=-12 and b=-15. Compute the distance |a-b| exactly.
G001— On the real number line, a=-12 and b=-15. Compute the distance |a-b| exactly.
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 · calculation
Before
a−b := sub(-12, -15)
After
a−b = 3
Conditions: none
Compute the signed difference first, not the distance itself.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S001
S003 · implies
Before
a−b=3
After
distance=|a−b|=3
Conditions: none
Use the sign branch for the difference; the resulting distance is nonnegative.
Step identifiers and dependencies
Rule: number_line_metric · Uses: G001, S002
S004 · calculation
Before
b−a := sub(-15, -12)
After
b−a = -3
Conditions: none
Swapping the points reverses the difference's sign.
Step identifiers and dependencies
Rule: exact_rational_arithmetic · Uses: G001, S003
S005 · implies
Before
|a−b|=3; |b−a|=3
After
swapping endpoints preserves distance
Conditions: none
Endpoint order changes but the absolute values agree, completing the reverse audit.
Step identifiers and dependencies
Rule: distance_symmetry · 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 b−a 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 — Recompute a−b with exact rational arithmetic, not rounded decimals.S001· premise_and_result_audit — Use ≥0 rather than >0 so that0 is not omitted.
Result
3
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: 424dd022232d0b8011d99f2332b3b40c22d0e965eb53a3941e883b003d2b1152candidate record SHA-256: e3d018c0585e14b89553b9a6454ae73f587ef74483922f8bcf5446c25bf0df37semantic review receipt SHA-256: c3b44ce8ed60ada6902fb206627537d5f90ba2a867eaba43a36f903280c496ccquestion-first witness SHA-256: 3e91602adc7f81a2671a2a3ffaf6a447028b56abc5853e8271673d76f30cd7adLean: NOT_RUN · human review: NOT_PERFORMED