Help me solve this linear inequality word problem?

Question

The sum of the lengths of any two sides of a triangle must be greater than the third side. If a triangle has one side that is 8 cm and a second side that is 4 cm less than twice the third side, what are the possible lengths for the second and third sides?

My head hurts now @_@

Answer 1

Let S and T be the second and third sides.

2T = S + 4.
S = 2T - 4.

OK. Our three inequalities are

T < 8 + S = 2T - 4.

I.e., T >4.

8 < S + T = 3T - 4.

That also tells us T>4.

S < 8 + T
2T - 4 < 8 + T
T < 12.

OK. It would seem that T can vary from 4 to 12 and, as it does, S = 2T - 4 varies from 4 to 20. At the endpoints of those intervals the triangle is either degenerate or not actually a triangle, depending on which way your teacher and textbook choose to define things. (Well, it's the latter since they said "greater" rather than "greater than or equal to" in the problem statement.)

Nothing funky is going on with S working out to less that 0 or something else weird, so that's your answer right there.

Answer 2

first side = 8
third side = x
second side = 2x - 4

8 + x > 2x - 4
8 + x - 2x > -4
8 - x > -4
-x > -4 - 8
-x > -12
x < 12

So...
now that you know the third side has to be less than 12, you can pick values for the third side and calculate the second side.