# Question 28631: Nonzero-Denominator Eligibility for an Average Rate

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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.

R0056-28631 · q-20261005-advance-20261006-r03-rate-of-change-0031



## Original question and givens

The proposed permitted endpoint inputs are x0=-5 and x1=-10. Is their difference a valid nonzero denominator for an average rate of change?


- G001 — The proposed permitted endpoint inputs are x0=-5 and x1=-10. Is their difference a valid nonzero denominator for an average rate of change?





## 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-rate-of-change](https://ocme.evemisslab.com/en/mko/mko-rate-of-change/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-rate-denominator-validity, denominator_validity]

After

derived = [distinct_endpoint_eligibility]

Conditions: none

Use the declared scalar/set/Yes-No/fraction answer role;integer fractions retain/1.Do not add an undeclared domain or change original bilingual data,formulas or questions.

Step identifiers and dependenciesRule: R56_source_answer_role_and_EXACT_family_interface · Uses: G001





S002 · calculationBefore

Rate ORIGINAL x1 minus x0 := sub(-10, -5)

After

Rate ORIGINAL x1 minus x0 = -5

Conditions: none

Perform exact rational arithmetic on the ORIGINAL roles.SAFE bounds raw inputs only;derived integers are unbounded and never approximated by floats.

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





S003 · impliesBefore

source roles = [-5, -10, -5]

After

derived = [Yes, Yes]

Conditions: none

The average rate requires x1-x0 nonzero.Equal endpoints are a legal No eligibility case;do not compute0/0 or repair the endpoints.

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









## Reverse checks


- S003 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.
- S002 · premise_and_result_audit — Recompute Rate ORIGINAL x1 minus x0 with exact rational arithmetic, not rounded decimals.
- S001 · premise_and_result_audit — Reverse-check EACH original datum and premise,retaining indices,signs and strict/non-strict boundaries.A correct final answer cannot replace the derivation obligations.This is candidate teaching/exact checking,not formal proof or publication clearance.





## Result

Yes

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: b04ddfcbe19c0afd4db4e45002e162e949490a1a71239a57657def8ed2ca850f
- candidate record SHA-256: 0f743907cf089cb3e5de2b4bc798c012fa38f037d927575c2691de30ce6cabfa
- semantic review receipt SHA-256: 8bbda4edbed7078634ae88342998c79bcfb95de96187fac119c3b08e0adb0bc8
- question-first witness SHA-256: 3edafa72d63d16a22414892b55806430e0a25b15b926d04f312550179607759f
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261005-advance-20261006-r03-rate-of-change-0031.json)