To find the possible values for [tex]\( n \)[/tex] in a triangle with side lengths [tex]\( 2x + 2 \, \text{ft} \)[/tex], [tex]\( x + 3 \, \text{ft} \)[/tex], and [tex]\( n \, \text{ft} \)[/tex], we need to apply 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 have three inequalities to consider:
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 step-by-step:
1. For the first inequality:
[tex]\[
(2x + 2) + (x + 3) > n
\][/tex]
[tex]\[
3x + 5 > n
\][/tex]
[tex]\[
n < 3x + 5
\][/tex]
2. For the second inequality:
[tex]\[
(2x + 2) + n > (x + 3)
\][/tex]
[tex]\[
2x + 2 + n > x + 3
\][/tex]
[tex]\[
2x + n + 2 > x + 3
\][/tex]
[tex]\[
2x + n > x + 1
\][/tex]
[tex]\[
n > x - 1
\][/tex]
3. For the third inequality:
[tex]\[
(x + 3) + n > (2x + 2)
\][/tex]
[tex]\[
x + n + 3 > 2x + 2
\][/tex]
[tex]\[
n + x + 3 > 2x + 2
\][/tex]
[tex]\[
n + 3 > x + 2
\][/tex]
[tex]\[
n > x - 1
\][/tex]
Notice that the result from the third inequality [tex]\( n > x - 1 \)[/tex] is the same as the result from the second inequality. Therefore, the combined simplified inequalities are:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Thus, we have concluded that the expression representing the possible values of [tex]\( n \)[/tex] is:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Therefore, the correct answer is:
[tex]\[
x-1
