To determine which expression represents the possible values of [tex]\( n \)[/tex] that form a valid triangle with sides [tex]\( 2x + 2 \)[/tex], [tex]\( x + 3 \)[/tex], and [tex]\( n \)[/tex] (all in feet), we must 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 third side.
Let's denote the sides of the triangle as follows:
- Side 1: [tex]\( a = 2x + 2 \)[/tex]
- Side 2: [tex]\( b = x + 3 \)[/tex]
- Side 3: [tex]\( c = n \)[/tex]
We will apply the triangle inequality theorem to obtain the possible values for [tex]\( n \)[/tex].
### Step 1: Apply the Inequality [tex]\( a + b > c \)[/tex]
[tex]\[ (2x + 2) + (x + 3) > n \][/tex]
Simplify:
[tex]\[ 3x + 5 > n \][/tex]
Thus:
[tex]\[ n < 3x + 5 \][/tex]
### Step 2: Apply the Inequality [tex]\( a + c > b \)[/tex]
[tex]\[ (2x + 2) + n > (x + 3) \][/tex]
Simplify:
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ x + 2 + n > 3 \][/tex]
Subtract 2 from both sides:
[tex]\[ x + n > 1 \][/tex]
Therefore:
[tex]\[ n > 1 - x \][/tex]
Since [tex]\( x + n > 1 \)[/tex] is valid in all cases where [tex]\( n > x - 1 \)[/tex], we derive:
[tex]\[ n > x - 1 \][/tex]
### Step 3: Apply the Inequality [tex]\( b + c > a \)[/tex]
[tex]\[ (x + 3) + n > (2x + 2) \][/tex]
Simplify:
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
Subtract [tex]\( x \)[/tex] from both sides:
[tex]\[ 3 + n > x + 2 \][/tex]
Subtract 2 from both sides:
[tex]\[ n > x - 1 \][/tex]
Thus, the same inequality is confirmed:
[tex]\[ n > x - 1 \][/tex]
### Conclusion
Combining the results of all three inequalities, we have:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
This can be written in interval notation:
[tex]\[ n \in (x - 1, 3x + 5) \][/tex]
Therefore, the expression that correctly represents the possible values of [tex]\( n \)[/tex] for the triangle is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]
