Final answer:
The restrictions on x for triangle ABC can be determined using the triangle inequality theorem, which involves solving three inequalities formed by the side lengths x+2, 2x, and 5x-8.
Step-by-step explanation:
To find the restrictions on x for the sides of triangle ABC, we must apply the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. In the case of triangle ABC with side lengths x+2, 2x, and 5x-8, we have three inequalities:
- x + 2 + 2x > 5x - 8 (Sum of two shorter sides is greater than the longest side)
- x + 2 + 5x - 8 > 2x (Sum of one shorter side and the longest side is greater than the middle side)
- 2x + 5x - 8 > x + 2 (Sum of the middle side and the longest side is greater than the shortest side)
By solving these inequalities, we can find the range of values for x in which a triangle with sides x+2, 2x, and 5x-8 can exist. The steps of this process would include things like combining like terms and isolating x on one side of the inequality.
However, the reference information provided is about completing the square for quadratic equations and subtracting x from both sides of an equation, which does not directly apply to this problem.