# Rational Absolute Branches, Targets, and Closed Thresholds

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261010-advance-20261014-r06-absolute-value-foundations · 2026-10-10 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-12/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-4/1, negative, 4/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-4/1, negative, 4/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0001/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 2. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=11/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-11/3, negative, 11/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-11/3, negative, 11/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0002/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 3. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-10/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-10/3, negative, 10/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-10/3, negative, 10/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0003/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 4. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=9/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-3/1, negative, 3/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-3/1, negative, 3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0004/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 5. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-8/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-8/3, negative, 8/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-8/3, negative, 8/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0005/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 6. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=7/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-7/3, negative, 7/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-7/3, negative, 7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0006/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 7. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-6/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-2/1, negative, 2/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-2/1, negative, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0007/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 8. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=5/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-5/3, negative, 5/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-5/3, negative, 5/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0008/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 9. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-4/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-4/3, negative, 4/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-4/3, negative, 4/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0009/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 10. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=3/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-1/1, negative, 1/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-1/1, negative, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0010/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 11. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-2/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-2/3, negative, 2/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-2/3, negative, 2/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0011/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 12. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=1/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(-1/3, negative, 1/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (-1/3, negative, 1/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0012/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 13. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=0/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(0/1, nonnegative, 0/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (0/1, nonnegative, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0013/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 14. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-1/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(1/3, nonnegative, 1/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (1/3, nonnegative, 1/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0014/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 15. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=2/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(2/3, nonnegative, 2/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (2/3, nonnegative, 2/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0015/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 16. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-3/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(1/1, nonnegative, 1/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (1/1, nonnegative, 1/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0016/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 17. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=4/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(4/3, nonnegative, 4/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (4/3, nonnegative, 4/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0017/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 18. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-5/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(5/3, nonnegative, 5/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (5/3, nonnegative, 5/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0018/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 19. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=6/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(2/1, nonnegative, 2/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (2/1, nonnegative, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0019/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 20. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-7/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(7/3, nonnegative, 7/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (7/3, nonnegative, 7/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0020/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 21. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=8/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(8/3, nonnegative, 8/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (8/3, nonnegative, 8/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0021/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 22. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-9/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(3/1, nonnegative, 3/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (3/1, nonnegative, 3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0022/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 23. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=10/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(10/3, nonnegative, 10/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (10/3, nonnegative, 10/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0023/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 24. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=-11/-3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(11/3, nonnegative, 11/3)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (11/3, nonnegative, 11/3).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0024/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 25. Normalized Rational Absolute-Value Branch

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. x=12/3. Give (reduced x, nonnegative or negative branch, abs(x)); zero belongs to the nonnegative branch.

View answer and explanation(4/1, nonnegative, 4/1)

Choose the branch by the normalized value’s sign; zero is not negative. A finite rational slice does not prove every real absolute-value property. The exact result, retaining the requested order, is (4/1, nonnegative, 4/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0025/)

Absolute value · qf-density14-absolute-rational-two-branches · candidate





## 26. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 0/1, 0/1]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(0/1, 0/1, 0/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (0/1, 0/1, 0/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0026/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 27. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 1/1, 0/1]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(1/1, 1/1, 0/1, 2/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (1/1, 1/1, 0/1, 2/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0027/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 28. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 2/2, 3/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(1/1, 1/2, 3/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (1/1, 1/2, 3/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0028/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 29. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 3/2, 4/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(3/2, 1/2, 2/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (3/2, 1/2, 2/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0029/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 30. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 4/2, 5/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(2/1, 1/2, 5/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (2/1, 1/2, 5/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0030/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 31. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 5/2, 6/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(5/2, 1/2, 3/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (5/2, 1/2, 3/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0031/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 32. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 6/2, 7/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(3/1, 1/2, 7/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (3/1, 1/2, 7/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0032/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 33. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 7/2, 8/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/2, 1/2, 4/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (7/2, 1/2, 4/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0033/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 34. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 8/2, 9/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(4/1, 1/2, 9/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (4/1, 1/2, 9/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0034/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 35. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 9/2, 10/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(9/2, 1/2, 5/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (9/2, 1/2, 5/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0035/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 36. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 10/2, 11/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(5/1, 1/2, 11/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (5/1, 1/2, 11/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0036/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 37. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 11/2, 12/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(11/2, 1/2, 6/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (11/2, 1/2, 6/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0037/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 38. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 12/2, 13/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(6/1, 1/2, 13/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (6/1, 1/2, 13/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0038/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 39. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 13/2, 14/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(13/2, 1/2, 7/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (13/2, 1/2, 7/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0039/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 40. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 14/2, 15/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(7/1, 1/2, 15/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (7/1, 1/2, 15/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0040/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 41. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 15/2, 16/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(15/2, 1/2, 8/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (15/2, 1/2, 8/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0041/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 42. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 16/2, 17/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(8/1, 1/2, 17/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (8/1, 1/2, 17/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0042/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 43. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 17/2, 18/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(17/2, 1/2, 9/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (17/2, 1/2, 9/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0043/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 44. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 18/2, 19/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(9/1, 1/2, 19/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (9/1, 1/2, 19/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0044/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 45. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 19/2, 20/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(19/2, 1/2, 10/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (19/2, 1/2, 10/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0045/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 46. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 20/2, 21/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(10/1, 1/2, 21/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (10/1, 1/2, 21/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0046/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 47. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 21/2, 22/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(21/2, 1/2, 11/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (21/2, 1/2, 11/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0047/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 48. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 22/2, 23/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(11/1, 1/2, 23/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (11/1, 1/2, 23/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0048/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 49. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 23/2, 24/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(23/2, 1/2, 12/1, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (23/2, 1/2, 12/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0049/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 50. Three Rational Distances and Path Defect

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Ordered points=[0/1, 24/2, 25/2]. Give (d01, d12, d02, d01+d12−d02). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(12/1, 1/2, 25/2, 0/1)

Take the absolute difference for each segment separately; a path sum is not an endpoint distance, and finite checks are not a universal triangle-inequality proof. The exact result, retaining the requested order, is (12/1, 1/2, 25/2, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0050/)

Absolute value · qf-density14-absolute-three-point-distances · candidate





## 51. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=0/1≥0, ordered points=[0/1, 1/1, -1/1, 0/-2]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(0/1, [0/1, 1/1, 1/1, 0/1], [Yes, No, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (0/1, [0/1, 1/1, 1/1, 0/1], [Yes, No, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0051/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 52. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=1/2≥0, ordered points=[-1/2, 1/2, 3/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(1/2, [1/2, 1/2, 3/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (1/2, [1/2, 1/2, 3/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0052/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 53. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=2/2≥0, ordered points=[-2/2, 2/2, 4/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(1/1, [1/1, 1/1, 2/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (1/1, [1/1, 1/1, 2/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0053/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 54. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=3/2≥0, ordered points=[-3/2, 3/2, 5/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(3/2, [3/2, 3/2, 5/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (3/2, [3/2, 3/2, 5/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0054/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 55. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=4/2≥0, ordered points=[-4/2, 4/2, 6/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(2/1, [2/1, 2/1, 3/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (2/1, [2/1, 2/1, 3/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0055/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 56. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=5/2≥0, ordered points=[-5/2, 5/2, 7/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(5/2, [5/2, 5/2, 7/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (5/2, [5/2, 5/2, 7/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0056/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 57. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=6/2≥0, ordered points=[-6/2, 6/2, 8/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(3/1, [3/1, 3/1, 4/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (3/1, [3/1, 3/1, 4/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0057/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 58. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=7/2≥0, ordered points=[-7/2, 7/2, 9/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(7/2, [7/2, 7/2, 9/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (7/2, [7/2, 7/2, 9/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0058/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 59. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=8/2≥0, ordered points=[-8/2, 8/2, 10/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(4/1, [4/1, 4/1, 5/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (4/1, [4/1, 4/1, 5/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0059/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 60. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=9/2≥0, ordered points=[-9/2, 9/2, 11/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(9/2, [9/2, 9/2, 11/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (9/2, [9/2, 9/2, 11/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0060/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 61. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=10/2≥0, ordered points=[-10/2, 10/2, 12/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(5/1, [5/1, 5/1, 6/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (5/1, [5/1, 5/1, 6/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0061/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 62. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=11/2≥0, ordered points=[-11/2, 11/2, 13/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(11/2, [11/2, 11/2, 13/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (11/2, [11/2, 11/2, 13/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0062/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 63. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=12/2≥0, ordered points=[-12/2, 12/2, 14/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(6/1, [6/1, 6/1, 7/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (6/1, [6/1, 6/1, 7/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0063/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 64. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=13/2≥0, ordered points=[-13/2, 13/2, 15/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(13/2, [13/2, 13/2, 15/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (13/2, [13/2, 13/2, 15/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0064/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 65. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=14/2≥0, ordered points=[-14/2, 14/2, 16/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(7/1, [7/1, 7/1, 8/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (7/1, [7/1, 7/1, 8/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0065/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 66. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=15/2≥0, ordered points=[-15/2, 15/2, 17/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(15/2, [15/2, 15/2, 17/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (15/2, [15/2, 15/2, 17/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0066/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 67. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=16/2≥0, ordered points=[-16/2, 16/2, 18/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(8/1, [8/1, 8/1, 9/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (8/1, [8/1, 8/1, 9/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0067/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 68. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=17/2≥0, ordered points=[-17/2, 17/2, 19/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(17/2, [17/2, 17/2, 19/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (17/2, [17/2, 17/2, 19/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0068/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 69. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=18/2≥0, ordered points=[-18/2, 18/2, 20/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(9/1, [9/1, 9/1, 10/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (9/1, [9/1, 9/1, 10/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0069/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 70. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=19/2≥0, ordered points=[-19/2, 19/2, 21/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(19/2, [19/2, 19/2, 21/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (19/2, [19/2, 19/2, 21/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0070/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 71. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=20/2≥0, ordered points=[-20/2, 20/2, 22/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(10/1, [10/1, 10/1, 11/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (10/1, [10/1, 10/1, 11/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0071/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 72. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=21/2≥0, ordered points=[-21/2, 21/2, 23/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(21/2, [21/2, 21/2, 23/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (21/2, [21/2, 21/2, 23/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0072/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 73. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=22/2≥0, ordered points=[-22/2, 22/2, 24/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(11/1, [11/1, 11/1, 12/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (11/1, [11/1, 11/1, 12/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0073/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 74. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=23/2≥0, ordered points=[-23/2, 23/2, 25/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(23/2, [23/2, 23/2, 25/2, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (23/2, [23/2, 23/2, 25/2, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0074/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 75. Inclusive Absolute-Value Threshold Mask

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. r=24/2≥0, ordered points=[-24/2, 24/2, 26/2, 0/1]. Give (r, four absolute values, four abs(point)≤r eligibility values), including endpoints.

View answer and explanation(12/1, [12/1, 12/1, 13/1, 0/1], [Yes, Yes, No, Yes])

Evaluate each absolute value, then compare with the exact closed threshold; r=0 is valid and finite substitutions are not general inequality solving. The exact result, retaining the requested order, is (12/1, [12/1, 12/1, 13/1, 0/1], [Yes, Yes, No, Yes]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0075/)

Absolute value · qf-density14-absolute-closed-radius-mask · candidate





## 76. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=-1/1, ordered queries=[0/1, 1/1, 2/1]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(No, [], [0/1, 1/1, 2/1], [No, No, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (No, [], [0/1, 1/1, 2/1], [No, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0076/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 77. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=0/1, ordered queries=[0/1, 1/1, -1/1]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [0/1], [0/1, 1/1, 1/1], [Yes, No, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [0/1], [0/1, 1/1, 1/1], [Yes, No, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0077/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 78. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=1/2, ordered queries=[-1/2, 1/2, 3/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-1/2, 1/2], [1/2, 1/2, 3/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-1/2, 1/2], [1/2, 1/2, 3/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0078/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 79. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=2/2, ordered queries=[-2/2, 2/2, 4/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-1/1, 1/1], [1/1, 1/1, 2/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-1/1, 1/1], [1/1, 1/1, 2/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0079/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 80. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=3/2, ordered queries=[-3/2, 3/2, 5/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-3/2, 3/2], [3/2, 3/2, 5/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-3/2, 3/2], [3/2, 3/2, 5/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0080/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 81. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=4/2, ordered queries=[-4/2, 4/2, 6/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-2/1, 2/1], [2/1, 2/1, 3/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-2/1, 2/1], [2/1, 2/1, 3/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0081/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 82. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=5/2, ordered queries=[-5/2, 5/2, 7/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-5/2, 5/2], [5/2, 5/2, 7/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-5/2, 5/2], [5/2, 5/2, 7/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0082/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 83. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=6/2, ordered queries=[-6/2, 6/2, 8/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-3/1, 3/1], [3/1, 3/1, 4/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-3/1, 3/1], [3/1, 3/1, 4/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0083/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 84. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=7/2, ordered queries=[-7/2, 7/2, 9/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-7/2, 7/2], [7/2, 7/2, 9/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-7/2, 7/2], [7/2, 7/2, 9/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0084/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 85. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=8/2, ordered queries=[-8/2, 8/2, 10/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-4/1, 4/1], [4/1, 4/1, 5/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-4/1, 4/1], [4/1, 4/1, 5/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0085/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 86. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=9/2, ordered queries=[-9/2, 9/2, 11/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-9/2, 9/2], [9/2, 9/2, 11/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-9/2, 9/2], [9/2, 9/2, 11/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0086/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 87. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=10/2, ordered queries=[-10/2, 10/2, 12/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-5/1, 5/1], [5/1, 5/1, 6/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-5/1, 5/1], [5/1, 5/1, 6/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0087/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 88. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=11/2, ordered queries=[-11/2, 11/2, 13/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-11/2, 11/2], [11/2, 11/2, 13/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-11/2, 11/2], [11/2, 11/2, 13/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0088/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 89. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=12/2, ordered queries=[-12/2, 12/2, 14/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-6/1, 6/1], [6/1, 6/1, 7/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-6/1, 6/1], [6/1, 6/1, 7/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0089/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 90. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=13/2, ordered queries=[-13/2, 13/2, 15/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-13/2, 13/2], [13/2, 13/2, 15/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-13/2, 13/2], [13/2, 13/2, 15/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0090/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 91. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=14/2, ordered queries=[-14/2, 14/2, 16/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-7/1, 7/1], [7/1, 7/1, 8/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-7/1, 7/1], [7/1, 7/1, 8/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0091/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 92. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=15/2, ordered queries=[-15/2, 15/2, 17/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-15/2, 15/2], [15/2, 15/2, 17/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-15/2, 15/2], [15/2, 15/2, 17/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0092/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 93. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=16/2, ordered queries=[-16/2, 16/2, 18/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-8/1, 8/1], [8/1, 8/1, 9/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-8/1, 8/1], [8/1, 8/1, 9/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0093/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 94. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=17/2, ordered queries=[-17/2, 17/2, 19/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-17/2, 17/2], [17/2, 17/2, 19/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-17/2, 17/2], [17/2, 17/2, 19/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0094/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 95. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=18/2, ordered queries=[-18/2, 18/2, 20/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-9/1, 9/1], [9/1, 9/1, 10/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-9/1, 9/1], [9/1, 9/1, 10/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0095/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 96. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=19/2, ordered queries=[-19/2, 19/2, 21/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-19/2, 19/2], [19/2, 19/2, 21/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-19/2, 19/2], [19/2, 19/2, 21/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0096/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 97. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=20/2, ordered queries=[-20/2, 20/2, 22/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-10/1, 10/1], [10/1, 10/1, 11/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-10/1, 10/1], [10/1, 10/1, 11/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0097/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 98. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=21/2, ordered queries=[-21/2, 21/2, 23/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-21/2, 21/2], [21/2, 21/2, 23/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-21/2, 21/2], [21/2, 21/2, 23/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0098/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 99. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=22/2, ordered queries=[-22/2, 22/2, 24/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-11/1, 11/1], [11/1, 11/1, 12/1], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-11/1, 11/1], [11/1, 11/1, 12/1], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0099/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate





## 100. Absolute-Value Target Roots and Candidate Audit

Use exact rational absolute values embedded in R; negative representation denominators are normalized before deciding signs. Target r=23/2, ordered queries=[-23/2, 23/2, 25/2]. Give (is r nonnegative?, increasing roots of |x|=r, three absolute query values, three equality-to-r values).

View answer and explanation(Yes, [-23/2, 23/2], [23/2, 23/2, 25/2], [Yes, Yes, No])

A negative target has no roots, zero has only root zero, and a positive target has two signed roots; substitute each candidate exactly. The exact result, retaining the requested order, is (Yes, [-23/2, 23/2], [23/2, 23/2, 25/2], [Yes, Yes, No]).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261010-advance-20261014-r06-absolute-value-0100/)

Absolute value · qf-density14-absolute-preimage-rational-audit · candidate



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