OCMEBilingual mathematics & derivations繁體中文

Rational Absolute Branches, Targets, and Closed Thresholds

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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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