Final answer:
The triangle inequality theorem, alongside the given ratios, indicates that the range of possible values for x to satisfy the conditions of the triangle CMB is x > 1.
Step-by-step explanation:
The student's question pertains to the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given the ratios TC:CM = 4:5 and AB:BC = 7:3, we can denote the side lengths as follows: TC = 4x, CM = 5x, AB = 7y, and BC = 3y.
We know that in any triangle, the following inequalities must be true:
- TC + CM > BM
- TC + BM > CM
- BM + CM > TC
In this case, since TC and CM are part of the same triangle CMB, one of the inequalities we can form is 4x + 5x > BM, meaning BM < 9x. Since AB and BC are also sides of a triangle with length BM, we can use the ratio 7y:3y to understand that BM = AB + BC = 7y + 3y = 10y. But for any value of x that would satisfy the triangle inequality, x must be greater than some positive value. Given that y is a positive value, we can deduce that x cannot simply be greater than zero but must be a larger positive value to satisfy the inequality BM < 9x, which leads us to conclude that x must be greater than 1.
Hence, the range of possible values for x to satisfy the triangle inequality in triangle CMB is x > 1.