To determine the possible values of [tex]\( n \)[/tex] for the side lengths [tex]\( 2x + 2 \)[/tex], [tex]\( x + 3 \)[/tex], and [tex]\( n \)[/tex] to form a triangle, we need to ensure they satisfy the triangle inequality theorem. The triangle inequality theorem states that for any triangle with sides [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
1. [tex]\( a + b > c \)[/tex]
2. [tex]\( a + c > b \)[/tex]
3. [tex]\( b + c > a \)[/tex]
Applying these inequalities to the given side lengths [tex]\( 2x + 2 \)[/tex], [tex]\( x + 3 \)[/tex], and [tex]\( n \)[/tex]:
1. [tex]\( (2x + 2) + (x + 3) > n \)[/tex]
[tex]\[ 3x + 5 > n \][/tex]
2. [tex]\( (2x + 2) + n > (x + 3) \)[/tex]
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
[tex]\[ x + n > 1 \][/tex]
[tex]\[ n > 1 - x \][/tex]
3. [tex]\( (x + 3) + n > (2x + 2) \)[/tex]
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
[tex]\[ n > x - 1 \][/tex]
Combining these inequalities:
[tex]\[ n > x - 1 \][/tex]
[tex]\[ 3x + 5 > n \][/tex]
Thus, the expression representing the possible values of [tex]\( n \)[/tex] is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{x - 1 < n < 3x + 5} \][/tex]
