linear inequalities

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 17 cm and a second side that is 1 cm less than twice the third side, what are the possible lengths for the second and third sides?

Answer 1

Let S be the third side. Notice that the second side is 2S-1. Then from the triangle inequality we get our system.
side 2 + side 3 > side 1
2S -1 + S > 17 (Move the 1 and combine the S's)
3S >18 (Divide both sides by 3)
S > 6
This gives us a lower bound for S

side 1 + side 3 > side 2
17 + S > 2S - 1 (Move the 1 and the S)
17 + 1 > 2S - S (Combine like terms)
18> S
Now we have an upper bound for S
Therefore 6<S<18
There is one last detail recall, second side = 2S-1, thus to find the possible values for that side you must plug in the lower and upper bounds for S