To determine the possible values of [tex]\( n \)[/tex] such that [tex]\( 2x+2 \)[/tex], [tex]\( x+3 \)[/tex], and [tex]\( n \)[/tex] can form a triangle, we must use the triangle inequality theorem. This 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 the sides of the triangle:
- Side 1: [tex]\( 2x + 2 \)[/tex]
- Side 2: [tex]\( x + 3 \)[/tex]
- Side 3: [tex]\( n \)[/tex]
We apply the triangle inequality theorem, which gives us three main 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]
Now we will simplify each inequality.
First Inequality:
[tex]\[
2x + 2 + x + 3 > n
\][/tex]
[tex]\[
3x + 5 > n
\][/tex]
[tex]\[
n < 3x + 5
\][/tex]
Second Inequality:
[tex]\[
2x + 2 + n > x + 3
\][/tex]
[tex]\[
2x + n + 2 > x + 3
\][/tex]
[tex]\[
2x - x + n + 2 > 3
\][/tex]
[tex]\[
x + n + 2 > 3
\][/tex]
[tex]\[
x + n > 1
\][/tex]
[tex]\[
n > 1 - x
\][/tex]
[tex]\[
n > x - 1
\][/tex]
Third Inequality:
[tex]\[
x + 3 + n > 2x + 2
\][/tex]
[tex]\[
n > 2x + 2 - x - 3
\][/tex]
[tex]\[
n > x - 1
\][/tex]
Combining all three inequalities:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Thus, the expression that represents the possible values of [tex]\( n \)[/tex] is:
[tex]\[
x - 1 < n < 3x + 5
\][/tex]
Therefore, the correct answer is [tex]\( \boxed{x - 1 < n < 3x + 5} \)[/tex].
