Final Answer:
The correct answer is (b) x + 17.
Step-by-step explanation:
To find the value of AB, we can use 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. In this case, AC represents one side of the triangle, and AB is the unknown side we are trying to find.
Given that AC = 39 and AB is represented by the expression x + 17, we can set up the inequality:
AC < AB + BC
39 < x + 17 + BC
Solving for BC, we get:
BC > 39 - (x + 17)
BC > 22 - x
Now, we know that BC is the remaining side of the triangle. To ensure the triangle is valid, BC must be greater than 0. So, we set up the inequality:
22 - x > 0
Solving for x, we find:
x < 22
Now, we substitute x + 17 for AB:
AB = x + 17
Since x < 22, the smallest possible value for AB is when x = 21, making AB equal to 38. Therefore, the correct answer is **(b) x + 17**, and it satisfies the triangle inequality theorem for all valid values of x.