Bunuel wrote:

If A, B, and C are distinct points on the number line. Is the length of the line segment AB less than 10?

(1) The sum of the lengths of line segments AC and BC is 10

(2) The sum of the lengths of line segments AB and AC is more than 10

We have 6 ways to order A, B and C.

1- A B C

2- A C B

3- B A C

4- B C A

5- C A B

6- C B A

We can say 5 and 6 are flip of 1 and 3 respectively but let's not make things complex.

Statement 1 - If sum of the lengths of line segments AC and BC is 10, then AB should be 10 or less in cases 1,3 5 and 6 while equal to 10 in 2 and 4. (Assume points 2, 6 and 12 on coordinate line and then check cases 1-6 above to understand the point) -

Not Sufficient.Statement 2 - If the sum of the lengths of line segments AB and AC is more than 10, then AB could theoretically be less or greater than 10 in any of the 6 cases. AB are closest in case 1, 3, 5 and 6 but if sum of AB and AC is greater than 10, AB could be less equal or greater than 10 (example case 1 (order of points A,B,C on number line) above, points 2,6,9 (AC+BC = 11 i.e. > 10 but AB < 10), points 2,12,13 (AB + BC = 21 i.e. > 10 but AB = 10) or points 2,13,14 (AB + AC = 23 i.e. > 10 but AB > 11) all illustrate this point-

Not Sufficient.Combined, since 2 doesn't add much info to 1, answer should be

E.

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