# Absolute Value Foundations

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-20260928-absolute-value-foundations · 2026-09-28 · 100 questions

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



## 1. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-12| exactly.

View answer and explanation12

-12<0, so absolute value takes the additive inverse and |-12|=12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0001/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 2. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-11| exactly.

View answer and explanation11

-11<0, so absolute value takes the additive inverse and |-11|=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0002/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 3. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-10| exactly.

View answer and explanation10

-10<0, so absolute value takes the additive inverse and |-10|=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0003/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 4. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-9| exactly.

View answer and explanation9

-9<0, so absolute value takes the additive inverse and |-9|=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0004/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 5. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-8| exactly.

View answer and explanation8

-8<0, so absolute value takes the additive inverse and |-8|=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0005/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 6. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-7| exactly.

View answer and explanation7

-7<0, so absolute value takes the additive inverse and |-7|=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0006/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 7. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-6| exactly.

View answer and explanation6

-6<0, so absolute value takes the additive inverse and |-6|=6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0007/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 8. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-5| exactly.

View answer and explanation5

-5<0, so absolute value takes the additive inverse and |-5|=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0008/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 9. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-4| exactly.

View answer and explanation4

-4<0, so absolute value takes the additive inverse and |-4|=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0009/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 10. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-3| exactly.

View answer and explanation3

-3<0, so absolute value takes the additive inverse and |-3|=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0010/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 11. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-2| exactly.

View answer and explanation2

-2<0, so absolute value takes the additive inverse and |-2|=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0011/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 12. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |-1| exactly.

View answer and explanation1

-1<0, so absolute value takes the additive inverse and |-1|=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0012/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 13. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |0| exactly.

View answer and explanation0

0≥0, so absolute value keeps the input and |0|=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0013/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 14. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |1| exactly.

View answer and explanation1

1≥0, so absolute value keeps the input and |1|=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0014/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 15. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |2| exactly.

View answer and explanation2

2≥0, so absolute value keeps the input and |2|=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0015/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 16. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |3| exactly.

View answer and explanation3

3≥0, so absolute value keeps the input and |3|=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0016/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 17. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |4| exactly.

View answer and explanation4

4≥0, so absolute value keeps the input and |4|=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0017/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 18. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |5| exactly.

View answer and explanation5

5≥0, so absolute value keeps the input and |5|=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0018/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 19. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |6| exactly.

View answer and explanation6

6≥0, so absolute value keeps the input and |6|=6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0019/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 20. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |7| exactly.

View answer and explanation7

7≥0, so absolute value keeps the input and |7|=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0020/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 21. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |8| exactly.

View answer and explanation8

8≥0, so absolute value keeps the input and |8|=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0021/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 22. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |9| exactly.

View answer and explanation9

9≥0, so absolute value keeps the input and |9|=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0022/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 23. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |10| exactly.

View answer and explanation10

10≥0, so absolute value keeps the input and |10|=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0023/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 24. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |11| exactly.

View answer and explanation11

11≥0, so absolute value keeps the input and |11|=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0024/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 25. Exact Absolute Value

Using the piecewise definition over the real numbers, compute |12| exactly.

View answer and explanation12

12≥0, so absolute value keeps the input and |12|=12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-exact-evaluation-0025/)

Absolute value · qf-absolute-value-exact-evaluation · candidate





## 26. Piecewise Branch of Absolute Value

For the real number x=12, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=12≥0, so use the nonnegative branch x and obtain |x|=12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0001/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 27. Piecewise Branch of Absolute Value

For the real number x=11, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=11≥0, so use the nonnegative branch x and obtain |x|=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0002/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 28. Piecewise Branch of Absolute Value

For the real number x=10, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=10≥0, so use the nonnegative branch x and obtain |x|=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0003/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 29. Piecewise Branch of Absolute Value

For the real number x=9, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=9≥0, so use the nonnegative branch x and obtain |x|=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0004/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 30. Piecewise Branch of Absolute Value

For the real number x=8, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=8≥0, so use the nonnegative branch x and obtain |x|=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0005/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 31. Piecewise Branch of Absolute Value

For the real number x=7, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=7≥0, so use the nonnegative branch x and obtain |x|=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0006/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 32. Piecewise Branch of Absolute Value

For the real number x=6, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=6≥0, so use the nonnegative branch x and obtain |x|=6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0007/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 33. Piecewise Branch of Absolute Value

For the real number x=5, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=5≥0, so use the nonnegative branch x and obtain |x|=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0008/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 34. Piecewise Branch of Absolute Value

For the real number x=4, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=4≥0, so use the nonnegative branch x and obtain |x|=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0009/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 35. Piecewise Branch of Absolute Value

For the real number x=3, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=3≥0, so use the nonnegative branch x and obtain |x|=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0010/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 36. Piecewise Branch of Absolute Value

For the real number x=2, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=2≥0, so use the nonnegative branch x and obtain |x|=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0011/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 37. Piecewise Branch of Absolute Value

For the real number x=1, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=1≥0, so use the nonnegative branch x and obtain |x|=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0012/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 38. Piecewise Branch of Absolute Value

For the real number x=0, which branch of the piecewise definition of |x| applies?

View answer and explanationnonnegative branch x

x=0≥0, so use the nonnegative branch x and obtain |x|=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0013/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 39. Piecewise Branch of Absolute Value

For the real number x=-1, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-1<0, so use the negative branch -x and obtain |x|=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0014/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 40. Piecewise Branch of Absolute Value

For the real number x=-2, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-2<0, so use the negative branch -x and obtain |x|=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0015/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 41. Piecewise Branch of Absolute Value

For the real number x=-3, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-3<0, so use the negative branch -x and obtain |x|=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0016/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 42. Piecewise Branch of Absolute Value

For the real number x=-4, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-4<0, so use the negative branch -x and obtain |x|=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0017/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 43. Piecewise Branch of Absolute Value

For the real number x=-5, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-5<0, so use the negative branch -x and obtain |x|=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0018/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 44. Piecewise Branch of Absolute Value

For the real number x=-6, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-6<0, so use the negative branch -x and obtain |x|=6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0019/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 45. Piecewise Branch of Absolute Value

For the real number x=-7, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-7<0, so use the negative branch -x and obtain |x|=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0020/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 46. Piecewise Branch of Absolute Value

For the real number x=-8, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-8<0, so use the negative branch -x and obtain |x|=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0021/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 47. Piecewise Branch of Absolute Value

For the real number x=-9, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-9<0, so use the negative branch -x and obtain |x|=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0022/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 48. Piecewise Branch of Absolute Value

For the real number x=-10, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-10<0, so use the negative branch -x and obtain |x|=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0023/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 49. Piecewise Branch of Absolute Value

For the real number x=-11, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-11<0, so use the negative branch -x and obtain |x|=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0024/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 50. Piecewise Branch of Absolute Value

For the real number x=-12, which branch of the piecewise definition of |x| applies?

View answer and explanationnegative branch -x

x=-12<0, so use the negative branch -x and obtain |x|=12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-piecewise-branch-0025/)

Absolute value · qf-absolute-value-piecewise-branch · candidate





## 51. Distance on the Real Number Line

On the real number line, a=-12 and b=-15. Compute the distance |a-b| exactly.

View answer and explanation3

|a-b|=|-12-(-15)|=|3|=3; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0001/)

Absolute value · qf-absolute-value-distance · candidate





## 52. Distance on the Real Number Line

On the real number line, a=-11 and b=-8. Compute the distance |a-b| exactly.

View answer and explanation3

|a-b|=|-11-(-8)|=|-3|=3; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0002/)

Absolute value · qf-absolute-value-distance · candidate





## 53. Distance on the Real Number Line

On the real number line, a=-10 and b=-1. Compute the distance |a-b| exactly.

View answer and explanation9

|a-b|=|-10-(-1)|=|-9|=9; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0003/)

Absolute value · qf-absolute-value-distance · candidate





## 54. Distance on the Real Number Line

On the real number line, a=-9 and b=6. Compute the distance |a-b| exactly.

View answer and explanation15

|a-b|=|-9-(6)|=|-15|=15; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0004/)

Absolute value · qf-absolute-value-distance · candidate





## 55. Distance on the Real Number Line

On the real number line, a=-8 and b=13. Compute the distance |a-b| exactly.

View answer and explanation21

|a-b|=|-8-(13)|=|-21|=21; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0005/)

Absolute value · qf-absolute-value-distance · candidate





## 56. Distance on the Real Number Line

On the real number line, a=-7 and b=-11. Compute the distance |a-b| exactly.

View answer and explanation4

|a-b|=|-7-(-11)|=|4|=4; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0006/)

Absolute value · qf-absolute-value-distance · candidate





## 57. Distance on the Real Number Line

On the real number line, a=-6 and b=-4. Compute the distance |a-b| exactly.

View answer and explanation2

|a-b|=|-6-(-4)|=|-2|=2; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0007/)

Absolute value · qf-absolute-value-distance · candidate





## 58. Distance on the Real Number Line

On the real number line, a=-5 and b=3. Compute the distance |a-b| exactly.

View answer and explanation8

|a-b|=|-5-(3)|=|-8|=8; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0008/)

Absolute value · qf-absolute-value-distance · candidate





## 59. Distance on the Real Number Line

On the real number line, a=-4 and b=10. Compute the distance |a-b| exactly.

View answer and explanation14

|a-b|=|-4-(10)|=|-14|=14; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0009/)

Absolute value · qf-absolute-value-distance · candidate





## 60. Distance on the Real Number Line

On the real number line, a=-3 and b=-14. Compute the distance |a-b| exactly.

View answer and explanation11

|a-b|=|-3-(-14)|=|11|=11; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0010/)

Absolute value · qf-absolute-value-distance · candidate





## 61. Distance on the Real Number Line

On the real number line, a=-2 and b=-7. Compute the distance |a-b| exactly.

View answer and explanation5

|a-b|=|-2-(-7)|=|5|=5; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0011/)

Absolute value · qf-absolute-value-distance · candidate





## 62. Distance on the Real Number Line

On the real number line, a=-1 and b=0. Compute the distance |a-b| exactly.

View answer and explanation1

|a-b|=|-1-(0)|=|-1|=1; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0012/)

Absolute value · qf-absolute-value-distance · candidate





## 63. Distance on the Real Number Line

On the real number line, a=0 and b=7. Compute the distance |a-b| exactly.

View answer and explanation7

|a-b|=|0-(7)|=|-7|=7; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0013/)

Absolute value · qf-absolute-value-distance · candidate





## 64. Distance on the Real Number Line

On the real number line, a=1 and b=14. Compute the distance |a-b| exactly.

View answer and explanation13

|a-b|=|1-(14)|=|-13|=13; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0014/)

Absolute value · qf-absolute-value-distance · candidate





## 65. Distance on the Real Number Line

On the real number line, a=2 and b=-10. Compute the distance |a-b| exactly.

View answer and explanation12

|a-b|=|2-(-10)|=|12|=12; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0015/)

Absolute value · qf-absolute-value-distance · candidate





## 66. Distance on the Real Number Line

On the real number line, a=3 and b=-3. Compute the distance |a-b| exactly.

View answer and explanation6

|a-b|=|3-(-3)|=|6|=6; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0016/)

Absolute value · qf-absolute-value-distance · candidate





## 67. Distance on the Real Number Line

On the real number line, a=4 and b=4. Compute the distance |a-b| exactly.

View answer and explanation0

|a-b|=|4-(4)|=|0|=0; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0017/)

Absolute value · qf-absolute-value-distance · candidate





## 68. Distance on the Real Number Line

On the real number line, a=5 and b=11. Compute the distance |a-b| exactly.

View answer and explanation6

|a-b|=|5-(11)|=|-6|=6; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0018/)

Absolute value · qf-absolute-value-distance · candidate





## 69. Distance on the Real Number Line

On the real number line, a=6 and b=-13. Compute the distance |a-b| exactly.

View answer and explanation19

|a-b|=|6-(-13)|=|19|=19; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0019/)

Absolute value · qf-absolute-value-distance · candidate





## 70. Distance on the Real Number Line

On the real number line, a=7 and b=-6. Compute the distance |a-b| exactly.

View answer and explanation13

|a-b|=|7-(-6)|=|13|=13; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0020/)

Absolute value · qf-absolute-value-distance · candidate





## 71. Distance on the Real Number Line

On the real number line, a=8 and b=1. Compute the distance |a-b| exactly.

View answer and explanation7

|a-b|=|8-(1)|=|7|=7; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0021/)

Absolute value · qf-absolute-value-distance · candidate





## 72. Distance on the Real Number Line

On the real number line, a=9 and b=8. Compute the distance |a-b| exactly.

View answer and explanation1

|a-b|=|9-(8)|=|1|=1; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0022/)

Absolute value · qf-absolute-value-distance · candidate





## 73. Distance on the Real Number Line

On the real number line, a=10 and b=15. Compute the distance |a-b| exactly.

View answer and explanation5

|a-b|=|10-(15)|=|-5|=5; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0023/)

Absolute value · qf-absolute-value-distance · candidate





## 74. Distance on the Real Number Line

On the real number line, a=11 and b=-9. Compute the distance |a-b| exactly.

View answer and explanation20

|a-b|=|11-(-9)|=|20|=20; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0024/)

Absolute value · qf-absolute-value-distance · candidate





## 75. Distance on the Real Number Line

On the real number line, a=12 and b=-2. Compute the distance |a-b| exactly.

View answer and explanation14

|a-b|=|12-(-2)|=|14|=14; exchanging the endpoints gives the same distance.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-distance-0025/)

Absolute value · qf-absolute-value-distance · candidate





## 76. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=-4.

View answer and explanationno solution

Absolute value is nonnegative and cannot equal -4, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0001/)

Absolute value · qf-absolute-value-equation-class · candidate





## 77. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=-3.

View answer and explanationno solution

Absolute value is nonnegative and cannot equal -3, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0002/)

Absolute value · qf-absolute-value-equation-class · candidate





## 78. Solution Classification of an Absolute-Value Equation

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

View answer and explanationno solution

Absolute value is nonnegative and cannot equal -2, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0003/)

Absolute value · qf-absolute-value-equation-class · candidate





## 79. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=-1.

View answer and explanationno solution

Absolute value is nonnegative and cannot equal -1, so there is no solution.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0004/)

Absolute value · qf-absolute-value-equation-class · candidate





## 80. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=0.

View answer and explanationunique solution x=0

|x|=0 holds exactly at x=0.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0005/)

Absolute value · qf-absolute-value-equation-class · candidate





## 81. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=1.

View answer and explanationtwo solutions x=-1, 1

The real numbers at distance 1 from 0 are x=-1 and x=1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0006/)

Absolute value · qf-absolute-value-equation-class · candidate





## 82. Solution Classification of an Absolute-Value Equation

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

View answer and explanationtwo solutions x=-2, 2

The real numbers at distance 2 from 0 are x=-2 and x=2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0007/)

Absolute value · qf-absolute-value-equation-class · candidate





## 83. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=3.

View answer and explanationtwo solutions x=-3, 3

The real numbers at distance 3 from 0 are x=-3 and x=3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0008/)

Absolute value · qf-absolute-value-equation-class · candidate





## 84. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=4.

View answer and explanationtwo solutions x=-4, 4

The real numbers at distance 4 from 0 are x=-4 and x=4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0009/)

Absolute value · qf-absolute-value-equation-class · candidate





## 85. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=5.

View answer and explanationtwo solutions x=-5, 5

The real numbers at distance 5 from 0 are x=-5 and x=5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0010/)

Absolute value · qf-absolute-value-equation-class · candidate





## 86. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=6.

View answer and explanationtwo solutions x=-6, 6

The real numbers at distance 6 from 0 are x=-6 and x=6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0011/)

Absolute value · qf-absolute-value-equation-class · candidate





## 87. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=7.

View answer and explanationtwo solutions x=-7, 7

The real numbers at distance 7 from 0 are x=-7 and x=7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0012/)

Absolute value · qf-absolute-value-equation-class · candidate





## 88. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=8.

View answer and explanationtwo solutions x=-8, 8

The real numbers at distance 8 from 0 are x=-8 and x=8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0013/)

Absolute value · qf-absolute-value-equation-class · candidate





## 89. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=9.

View answer and explanationtwo solutions x=-9, 9

The real numbers at distance 9 from 0 are x=-9 and x=9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0014/)

Absolute value · qf-absolute-value-equation-class · candidate





## 90. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=10.

View answer and explanationtwo solutions x=-10, 10

The real numbers at distance 10 from 0 are x=-10 and x=10.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0015/)

Absolute value · qf-absolute-value-equation-class · candidate





## 91. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=11.

View answer and explanationtwo solutions x=-11, 11

The real numbers at distance 11 from 0 are x=-11 and x=11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0016/)

Absolute value · qf-absolute-value-equation-class · candidate





## 92. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=12.

View answer and explanationtwo solutions x=-12, 12

The real numbers at distance 12 from 0 are x=-12 and x=12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0017/)

Absolute value · qf-absolute-value-equation-class · candidate





## 93. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=13.

View answer and explanationtwo solutions x=-13, 13

The real numbers at distance 13 from 0 are x=-13 and x=13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0018/)

Absolute value · qf-absolute-value-equation-class · candidate





## 94. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=14.

View answer and explanationtwo solutions x=-14, 14

The real numbers at distance 14 from 0 are x=-14 and x=14.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0019/)

Absolute value · qf-absolute-value-equation-class · candidate





## 95. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=15.

View answer and explanationtwo solutions x=-15, 15

The real numbers at distance 15 from 0 are x=-15 and x=15.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0020/)

Absolute value · qf-absolute-value-equation-class · candidate





## 96. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=16.

View answer and explanationtwo solutions x=-16, 16

The real numbers at distance 16 from 0 are x=-16 and x=16.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0021/)

Absolute value · qf-absolute-value-equation-class · candidate





## 97. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=17.

View answer and explanationtwo solutions x=-17, 17

The real numbers at distance 17 from 0 are x=-17 and x=17.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0022/)

Absolute value · qf-absolute-value-equation-class · candidate





## 98. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=18.

View answer and explanationtwo solutions x=-18, 18

The real numbers at distance 18 from 0 are x=-18 and x=18.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0023/)

Absolute value · qf-absolute-value-equation-class · candidate





## 99. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=19.

View answer and explanationtwo solutions x=-19, 19

The real numbers at distance 19 from 0 are x=-19 and x=19.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0024/)

Absolute value · qf-absolute-value-equation-class · candidate





## 100. Solution Classification of an Absolute-Value Equation

Over the real numbers, classify the solution set of |x|=20.

View answer and explanationtwo solutions x=-20, 20

The real numbers at distance 20 from 0 are x=-20 and x=20.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20260928-absolute-value-equation-class-0025/)

Absolute value · qf-absolute-value-equation-class · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20260928-absolute-value-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20260928-absolute-value-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20260928-absolute-value-foundations/validation-report.json)