Final answer:
To find the range of possible x-values in a triangle, apply the triangle inequality theorem and solve the resulting inequalities.
Step-by-step explanation:
To find the range of possible x-values in a triangle with sides MN=x-1, NP=9x-68, and MP=5x-4, we need to determine the allowed regions for x.
Using the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
By applying this theorem to our given sides, we can set up the following inequalities:
- x-1 + 9x-68 > 5x-4
- x-1 + 5x-4 > 9x-68
- 5x-4 + 9x-68 > x-1
Simplifying each inequality, we get:
- 10x-69 > 5x-4
- 6x-5 > 9x-68
- 14x-72 > x-1
Solving each inequality, we find that x can range from approximately -2.67 to 18.18.