To solve this problem, we need to determine the minimum value of \( n \) such that the perimeter of the triangle is at least 37 units.
Given the side lengths of the triangle:
- First side: \( n \)
- Second side: \( n - 3 \)
- Third side: \( 2(n - 2) \)
The perimeter \( P \) of the triangle is the sum of the lengths of its sides. Therefore, we can express the perimeter as:
[tex]\[
P = n + (n - 3) + 2(n - 2)
\][/tex]
Simplify the expression:
[tex]\[
P = n + n - 3 + 2n - 4
\][/tex]
Combine like terms:
[tex]\[
P = 4n - 7
\][/tex]
We want the perimeter to be at least 37 units, so we set up the inequality:
[tex]\[
4n - 7 \geq 37
\][/tex]
To solve for \( n \), first isolate \( n \) on one side of the inequality:
[tex]\[
4n - 7 \geq 37
\][/tex]
Add 7 to both sides:
[tex]\[
4n \geq 44
\][/tex]
Divide both sides by 4:
[tex]\[
n \geq 11
\][/tex]
Therefore, the minimum value of \( n \) that satisfies this condition is \( 11 \). Thus, the correct answer is:
B. [tex]\( n \geq 11 \)[/tex]
