To determine the possible values of [tex]\(n\)[/tex] given the side lengths [tex]\(2x + 2\)[/tex] feet, [tex]\(x + 3\)[/tex] feet, and [tex]\(n\)[/tex] feet, we need to use the triangle inequality theorem. The 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.
We will use the triangle inequality theorem to set up the inequalities:
1. [tex]\(2x + 2 + (x + 3) > n\)[/tex]
2. [tex]\(2x + 2 + n > x + 3\)[/tex]
3. [tex]\(x + 3 + n > 2x + 2\)[/tex]
Let's solve each inequality one by one.
### Inequality 1: [tex]\(2x + 2 + (x + 3) > n\)[/tex]
Combine like terms:
[tex]\[2x + 2 + x + 3 > n\][/tex]
[tex]\[3x + 5 > n\][/tex]
[tex]\[n < 3x + 5\][/tex]
### Inequality 2: [tex]\(2x + 2 + n > x + 3\)[/tex]
Combine like terms:
[tex]\[2x + 2 + n > x + 3\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[2x + n + 2 > x + 3\][/tex]
[tex]\[x + n + 2 > 3\][/tex]
Subtract 2 from both sides:
[tex]\[n + x > 1\][/tex]
[tex]\[n > x - 1\][/tex]
### Inequality 3: [tex]\(x + 3 + n > 2x + 2\)[/tex]
Combine like terms:
[tex]\[x + 3 + n > 2x + 2\][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[3 + n > x + 2\][/tex]
Subtract 2 from both sides:
[tex]\[n + 1 > x\][/tex]
[tex]\[n > x - 1\][/tex]
This inequality [tex]\(n > x - 1\)[/tex] confirms the result from inequality 2 and does not provide additional constraints.
### Conclusion
Combining the valid inequalities from Inequality 1 and Inequality 2, we get:
[tex]\[x - 1 < n < 3x + 5\][/tex]
Thus, the expression that represents the possible values of [tex]\(n\)[/tex], in feet, is:
[tex]\[x - 1 < n < 3x + 5\][/tex]
The correct answer is:
[tex]\[x - 1 < n < 3x + 5\][/tex]
