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 in which the length of the third side of a triangle must lie, 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 remaining side.

Given:
- One side of the triangle is 29 units.
- Another side of the triangle is 40 units.
- Let [tex]\( x \)[/tex] be the length of the third side.

We apply the triangle inequality theorem to form the following inequalities:

1. The sum of the first two sides should be greater than the third side:
- [tex]\( 29 + 40 > x \)[/tex]
- This simplifies to [tex]\( 69 > x \)[/tex] or [tex]\( x < 69 \)[/tex].

2. The sum of one side and the third side should be greater than the second side:
- [tex]\( 29 + x > 40 \)[/tex]
- This simplifies to [tex]\( x > 11 \)[/tex].

3. The sum of the second side and the third side should be greater than the first side:
- [tex]\( 40 + x > 29 \)[/tex]
- This simplifies to [tex]\( x > -11 \)[/tex].

Since the condition [tex]\( x > 11 \)[/tex] is stronger than [tex]\( x > -11 \)[/tex], we use [tex]\( x > 11 \)[/tex].

Combining these inequalities, we get:
- [tex]\( 11 < x < 69 \)[/tex]

Therefore, the length of the third side must be greater than 11 units and less than 69 units. The correct answer is:

C. [tex]\( 11 < x < 69 \)[/tex]