Answer :
In order to determine the possible values of [tex]\( n \)[/tex] in the triangle, 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 remaining side. Let's apply this to the given triangle with sides [tex]\( 2x + 2 \)[/tex] feet, [tex]\( x + 3 \)[/tex] feet, and [tex]\( n \)[/tex] feet.
### Inequality 1
First, consider the inequality involving the sides [tex]\( 2x + 2 \)[/tex] and [tex]\( x + 3 \)[/tex]:
[tex]\[ (2x + 2) + (x + 3) > n \][/tex]
Simplify the left-hand side:
[tex]\[ 3x + 5 > n \][/tex]
Thus, we get:
[tex]\[ n < 3x + 5 \][/tex]
### Inequality 2
Next, consider the inequality involving the sides [tex]\( 2x + 2 \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ (2x + 2) + n > x + 3 \][/tex]
Simplify this inequality:
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
Subtract [tex]\( x + 2 \)[/tex] from both sides:
[tex]\[ x + n > 1 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ n > 1 - x \][/tex]
### Inequality 3
Finally, consider the inequality involving the sides [tex]\( x + 3 \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ (x + 3) + n > 2x + 2 \][/tex]
Simplify this inequality:
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
Subtract [tex]\( x + 2 \)[/tex] from both sides:
[tex]\[ n + 1 > x \][/tex]
Subtract 1 from both sides:
[tex]\[ n > x - 1 \][/tex]
Putting these inequalities together, we have:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
Therefore, the expression representing the possible values of [tex]\( n \)[/tex] in feet is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
This is the correct option among the given choices.
### Inequality 1
First, consider the inequality involving the sides [tex]\( 2x + 2 \)[/tex] and [tex]\( x + 3 \)[/tex]:
[tex]\[ (2x + 2) + (x + 3) > n \][/tex]
Simplify the left-hand side:
[tex]\[ 3x + 5 > n \][/tex]
Thus, we get:
[tex]\[ n < 3x + 5 \][/tex]
### Inequality 2
Next, consider the inequality involving the sides [tex]\( 2x + 2 \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ (2x + 2) + n > x + 3 \][/tex]
Simplify this inequality:
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
Subtract [tex]\( x + 2 \)[/tex] from both sides:
[tex]\[ x + n > 1 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ n > 1 - x \][/tex]
### Inequality 3
Finally, consider the inequality involving the sides [tex]\( x + 3 \)[/tex] and [tex]\( n \)[/tex]:
[tex]\[ (x + 3) + n > 2x + 2 \][/tex]
Simplify this inequality:
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
Subtract [tex]\( x + 2 \)[/tex] from both sides:
[tex]\[ n + 1 > x \][/tex]
Subtract 1 from both sides:
[tex]\[ n > x - 1 \][/tex]
Putting these inequalities together, we have:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
Therefore, the expression representing the possible values of [tex]\( n \)[/tex] in feet is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
This is the correct option among the given choices.
