Select the correct answer.

A triangle has one side of length 29 units and another of length 40 units. Determine the range in which the length of the third side must lie.

A. [tex]-11\ \textless \ x\ \textless \ 69[/tex]
B. [tex]11 \leq x \leq 69[/tex]
C. [tex]11\ \textless \ x\ \textless \ 69[/tex]
D. [tex]-11 \leq x \leq 69[/tex]

To determine the range for the length of the third side of a triangle when the first two sides are given as 29 units and 40 units, we can use 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.

Let's break it down step-by-step:

1. Identifying the given sides:
- First side, [tex]$ a = 29 $[/tex]
- Second side, [tex]$ b = 40 $[/tex]

2. Calculating the lower bound:
- According to the triangle inequality theorem, the difference between any two sides must be less than the third side.
- In this case, the difference between 29 and 40 is [tex]$ |29 - 40| = 11 $[/tex].
- Therefore, the third side must be greater than 11 units.

3. Calculating the upper bound:
- Again, according to the theorem, the sum of any two sides must be greater than the third side.
- The sum of 29 and 40 is [tex]$ 29 + 40 = 69 $[/tex].
- Therefore, the third side must be less than 69 units.

4. Combining the results:
- The third side (let’s call it [tex]$ x $[/tex]) must satisfy both conditions. It must be greater than 11 and less than 69.

Based on this reasoning, the range of the length of the third side is [tex]$ 11 < x < 69 $[/tex].

Thus, the correct answer is:
C. [tex]$ 11 < x < 69 $[/tex]