Let's solve the problem step-by-step.
Given the sides of the triangle:
- First side: [tex]\( n \)[/tex]
- Second side: [tex]\( n - 3 \)[/tex]
- Third side: [tex]\( 2(n - 2) \)[/tex]
First, let's simplify the expression for the third side:
[tex]\[ 2(n - 2) = 2n - 4 \][/tex]
The perimeter of the triangle is the sum of all three sides:
[tex]\[ \text{Perimeter} = n + (n - 3) + (2n - 4) \][/tex]
Now combine the terms:
[tex]\[ \text{Perimeter} = n + n - 3 + 2n - 4 \][/tex]
[tex]\[ \text{Perimeter} = 4n - 7 \][/tex]
We are given that the perimeter should be at least 37 units:
[tex]\[ 4n - 7 \geq 37 \][/tex]
To solve for [tex]\( n \)[/tex], first add 7 to both sides to isolate the term with [tex]\( n \)[/tex]:
[tex]\[ 4n - 7 + 7 \geq 37 + 7 \][/tex]
[tex]\[ 4n \geq 44 \][/tex]
Next, divide both sides by 4:
[tex]\[ n \geq \frac{44}{4} \][/tex]
[tex]\[ n \geq 11 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{n \geq 11} \][/tex]
So the correct option is D.
