Two sides of a triangle are 4 and 10..

[Macaroni]

This problem is from the Geometry Strategy Guide, Chapter 2, problem #7:

Two sides of a triangle are 4 and 10. If the third side is integer x, how many possible values are there for integer x?

Answer: 7

If two sides of a triangle are 4 and 10, the third side must be greater than 10 - 4 and smaller than 10 + 4. Therefore, the possible values for x are {7, 8, 9, 10, 11, 12, 13}.

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My Question:

I rationally understand that when I list the values out from 6 to 14, that there are 7 integer values, but I got the question wrong because I did 14 - 6 = 8.

How do I conceptually understand this? Am I not accounting for some kind of double counting?

Thank-you for your help in advance!

[RonPurewal]

Here's how I handle this sort of thing:

"- If there aren't a ton of values, I'll either list them (on paper) or literally count them on my fingers.
Don't be too proud to do these things. There are no points for style or for "elegant" solutions.

"- If there are too many values to count, I'll subtract something that makes the lower (starting) number into "1".
For instance, if the integers are 106 through 299, I'll subtract 105... so there are exactly as many integers as there are from 1 to 194. In other words, 194 integers.

[RonPurewal]

In case you're asking about WHY straight subtraction doesn't work (a consideration that's ultimately immaterial, but at least interesting)"”"”
When you subtract, you're not finding a count value; you're finding a distance between the two numbers. Those aren't the same; the distance will be 1 less than the number of "mileposts" (= integers).

If you only have 2 integers (e.g., 8 and 9), this is trivially easy to see. There are 2 integers, but the distance between them (= what you get by subtracting) is only 1 unit.
From there, adding more integers increases both the distance and the count value by 1 for each integer added. So, they will always be 1 apart.

[Macaroni]

Oooh okay, I get it. Thank-you!!

[RonPurewal]

Oooh okay, I get it. Thank-you!!

You're welcome.