To determine the possible values for [tex]\(n\)[/tex] in the context of a triangle with side lengths [tex]\(2x + 2\)[/tex] feet, [tex]\(x + 3\)[/tex] feet, and [tex]\(n\)[/tex] feet, we need to apply the triangle inequality theorem. The theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, we have to satisfy the following three 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]
First, let's simplify each inequality step by step.
### Inequality 1:
[tex]\[ (2x + 2) + (x + 3) > n \][/tex]
[tex]\[ 2x + x + 2 + 3 > n \][/tex]
[tex]\[ 3x + 5 > n \][/tex]
[tex]\[ n < 3x + 5 \][/tex]
### Inequality 2:
[tex]\[ (2x + 2) + n > (x + 3) \][/tex]
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
[tex]\[ 2x + n + 2 > x + 3 \][/tex]
Subtract [tex]\(x\)[/tex] from both sides:
[tex]\[ x + n + 2 > 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ n > x + 1 \][/tex]
### Inequality 3:
[tex]\[ (x + 3) + n > (2x + 2) \][/tex]
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
Subtract [tex]\(2x\)[/tex] from both sides:
[tex]\[ x + n + 3 > 2 \][/tex]
[tex]\[ n + 3 > x + 2 \][/tex]
Subtract 3 from both sides:
[tex]\[ n > x - 1 \][/tex]
Now, let’s combine the results of the inequalities:
- From Inequality 1: [tex]\(n < 3x + 5\)[/tex]
- From Inequality 2: [tex]\(n > x + 1\)[/tex]
- From Inequality 3: [tex]\(n > x - 1\)[/tex]
We need to take the most restrictive lower bound and the least restrictive upper bound:
- The lower bound is [tex]\(n > x - 1\)[/tex]
- The upper bound is [tex]\(n < 3x + 5\)[/tex]
Therefore, the expression representing the possible values of [tex]\(n\)[/tex] in feet is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
So, the correct answer is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
