To determine the possible values of [tex]\( n \)[/tex] for the given triangle side lengths, we need to 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 third side.
Given the sides:
1. [tex]\( 2x + 2 \)[/tex]
2. [tex]\( x + 3 \)[/tex]
3. [tex]\( n \)[/tex]
We need to set up and solve 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 work through each inequality step-by-step.
### Inequality 1:
[tex]\[
(2x + 2) + (x + 3) > n
\][/tex]
Combine the terms:
[tex]\[
3x + 5 > n
\][/tex]
Rewriting this, we get:
[tex]\[
n < 3x + 5
\][/tex]
### Inequality 2:
[tex]\[
(2x + 2) + n > (x + 3)
\][/tex]
Combine the terms:
[tex]\[
2x + 2 + n > x + 3
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
x + 2 + n > 3
\][/tex]
Subtract 2 from both sides:
[tex]\[
x + n > 1
\][/tex]
Rewriting this, we get:
[tex]\[
n > 1 - x
\][/tex]
### Inequality 3:
[tex]\[
(x + 3) + n > (2x + 2)
\][/tex]
Combine the terms:
[tex]\[
x + 3 + n > 2x + 2
\][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[
3 + n > x + 2
\][/tex]
Subtract 3 from both sides:
[tex]\[
n > x - 1
\][/tex]
### Combining the inequalities:
We have:
1. [tex]\( n < 3x + 5 \)[/tex]
2. [tex]\( n > x - 1 \)[/tex]
Therefore, the possible values for [tex]\( n \)[/tex] are:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Hence, the correct expression representing the possible values of [tex]\( n \)[/tex], in feet, is:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
