Let's analyze the problem step by step, applying the triangle inequality theorem. The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.
Given:
- Side 1: [tex]\( 2x + 2 \)[/tex]
- Side 2: [tex]\( x + 3 \)[/tex]
- Side 3: [tex]\( n \)[/tex]
We need to establish the conditions under which these three sides can form a triangle, focusing on [tex]\( n \)[/tex] as follows:
1. First Inequality: [tex]\( n \)[/tex] must be greater than the difference of any two sides.
- Consider [tex]\( n \)[/tex] compared to [tex]\( 2x + 2 \)[/tex] and [tex]\( x + 3 \)[/tex].
- [tex]\( n \)[/tex] must be greater than the difference of these two sides:
[tex]\[
n > |(2x + 2) - (x + 3)| \implies n > |x - 1|
\][/tex]
Since [tex]\( x \)[/tex] is a positive number, [tex]\( n > x - 1 \)[/tex].
2. Second Inequality: [tex]\( n \)[/tex] must be less than the sum of the other two sides.
- Consider [tex]\( n \)[/tex] compared to [tex]\( 2x + 2 \)[/tex] and [tex]\( x + 3 \)[/tex].
- [tex]\( n \)[/tex] must be less than the sum of these two sides:
[tex]\[
n < (2x + 2) + (x + 3) \implies n < 3x + 5
\][/tex]
Combining these two inequalities, we get:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Therefore, the expression that represents the possible values of [tex]\( n \)[/tex], in feet, is:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Thus, the correct answer is:
[tex]\[
\boxed{x - 1 < n < 3x + 5}
\][/tex]
