A Triangle Inequality and Algebra Application

Students learn the triangle inequality in geometry and might even recall it correctly 5 minutes after the final exam ("Let's see -- I think it's something like 'two sides of a triangle are more or is it less than the other side'"). Of course, if they can visualize it, they might retain it longer, but, in the end, they should know it as well as they know their basic arithmetic facts. (Uh oh -- bad analogy!).

Here's a fairly straightforward application although many students will answer it incorrectly even if they correctly recall the key geometric fact. Can you see where the traps lie?

A triangle has sides of lengths 12-x, 12 and 12+x. How many integer values of x are possible?


Now for the generalization:

A triangle has sides of lengths a-x, a and a+x, where a is a positive even integer. In terms of a, how many integer values of x are possible?


Comments:
(1) Do you believe we should more heavily emphasize the triangle inequality? The SAT certainly does (not that that is any justification, right?).
(2) What teaching strategies have worked best for you in the classroom when introducing this theorem? How much time is generally devoted to this topic?

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