# Question 1082: Solution Classification of an Absolute-Value Equation

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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.

R0001-01082 · q-20260928-absolute-value-equation-class-0007



## Original question and givens

Over the real numbers, classify the solution set of |x|=2.


- G001 — Over the real numbers, classify the solution set of |x|=2.





## 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|=2

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 dependenciesRule: absolute_nonnegativity · Uses: G001, S001





S003 · calculationBefore

negative candidate := mul(-1, 2)

After

negative candidate = -2

Conditions: none

In the negative branch, −x=c gives x=−c.

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





S004 · impliesBefore

x≥0 gives x=2; x<0 would give x=-2

After

distinct permitted roots=[-2, 2]

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 dependenciesRule: exhaustive_sign_cases · Uses: G001, S003





S005 · impliesBefore

[-2, 2]

After

each displayed root has |x|=2

Conditions: none

Substitute every candidate and check its branch sign to obtain every and only solution.

Step identifiers and dependenciesRule: 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=-2, 2

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: bf71ff504fab640814d4d48cccf51c9a93685b8e0c04738c4bf9f1c8b7cf7ee4
- candidate record SHA-256: 803c0a9a76b2990d1c9ea323a57c69227614a1584645c17dedf079a576cfa6c2
- 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-equation-class-0007.json)