Final answer:
The range of possible values for x in triangle ABC, taking into account the Triangle Inequality Theorem and ensuring the side lengths are positive, is 2 < x < 11.
Step-by-step explanation:
Determining the Possible Values for x in a Triangle
To determine the possible values for x in triangle ABC with side lengths 15, 3, and 2x - 4, we must use the Triangle Inequality Theorem. The Triangle Inequality 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. Therefore, we have two inequalities to consider:
- 15 + 3 > 2x - 4
- 15 + (2x - 4) > 3
To find the range of possible values for x, we will solve each inequality:
-
- 15 + 3 > 2x - 4
18 > 2x - 4
22 > 2x
11 > x
-
- 15 + 2x - 4 > 3
2x + 11 > 3
2x > -8
x > -4
Combining these inequalities, we find that the possible values for x are:
-4 < x < 11.
However, we must consider the third side which is not in the inequality. For x to be positive and the side to have a positive length, x must be greater than 2. Therefore, the final possible range for x, considering all sides of the triangle, is:
2 < x < 11.