Let's review the key property required for three lengths to form a triangle: the triangle inequality theorem. According to this theorem, for three sides of lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] to form a triangle, the following inequalities must hold:
1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]
In this question, we are given a triangle with side lengths [tex]\(2x + 2\)[/tex] feet, [tex]\(x + 3\)[/tex] feet, and [tex]\(n\)[/tex] feet. We need to determine the possible values of [tex]\(n\)[/tex] that satisfy the triangle inequality theorem.
Let's apply the inequalities one by one:
1. [tex]\( (2x + 2) + (x + 3) > n \)[/tex]
[tex]\[
2x + 2 + x + 3 > n \\
3x + 5 > n \\
n < 3x + 5
\][/tex]
2. [tex]\( (2x + 2) + n > (x + 3) \)[/tex]
[tex]\[
2x + 2 + n > x + 3 \\
2x + 2 + n - x > 3 \\
x + 2 + n > 3 \\
n > x + 1 - 2 \\
n > x + 1
\][/tex]
3. [tex]\( (x + 3) + n > (2x + 2) \)[/tex]
[tex]\[
x + 3 + n > 2x + 2 \\
n > 2x + 2 - x - 3 \\
n > x - 1
\][/tex]
By combining these results, we get the range:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Therefore, the expression that represents the possible values of [tex]\(n\)[/tex] is:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
So the correct answer is:
[tex]\[
\boxed{x - 1 < n < 3x + 5}
\][/tex]
