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 2323 cm and a second side that is 44 cm less than twice the third side, what are the possible lengths for the second and third sides?

Let's begin by identifying the lengths of the three sides of the triangle:    length of side 1 = 17    length of side 2 = 2x - 1 (1 less than twice side 3)    length of side 3 = x Now let's apply the Triangle Inequality Theorem to this triangle:    side 1 + side 2 > side 3:        17 + 2x - 1 > x        16 + 2x > x        2x - x > -16        x > -16 (reject negative measurement)        2x - 1 > -33 (reject negative measurement)     side 1 + side 3 > side 2        17 + x > 2x - 1        x - 2x > -1 - 17        -x > -18          x      < 18        2x - 1 < 35    side 2 + side 3 > side 1        2x - 1 + x > 17        3x - 1 > 17        3x > 18          x       >   6        2x - 1 > 11 Thus, we have our answers based on the value of x:      6 <   x       (length of side 3) < 18    11 < 2x - 1 (length of side 2) < 35 

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