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

1st Side: 8

2nd Side: 2x - 1

3rd Side: x

(8) + (2x - 1) > x (2x - 1) + x > 8 (x) + (8) > 2x - 1

2x + 7 > x 3x - 1 > 8 8 > x - 1

2x > - 7 3x > 9 9 > x

x > - 3.5 (disregard) x > 3 x < 9

3rd Side: x ⇒ (3 < x < 9)

2nd Side: 2x - 1 ⇒ 2 (3) - 1 < x < 2 (9) - 1 ⇒ 5 < x < 17

Answer: the 3rd side must be between 3 and 9, the 2nd side must be between 5 and 17