A triangle has side lengths measuring [tex]$2x + 2[tex]$[/tex] ft, [tex]$[/tex]x + 3[tex]$[/tex] ft, and [tex]n$[/tex][/tex] ft. Which expression represents the possible values of [tex]n[/tex] in feet? Express your answer in simplest terms.

A. [tex]x - 1 \ \textless \ n \ \textless \ 3x + 5[/tex]
B. [tex]n = 3x + 5[/tex]
C. [tex]n = x - 1[/tex]
D. [tex]3x + 5 \ \textless \ n \ \textless \ x - 1[/tex]



Answer :

To determine the valid expression for the possible values of \( n \) for the sides of a triangle, we can use the triangle inequality theorem. The triangle inequality theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\):

1. \( a + b > c \)
2. \( a + c > b \)
3. \( b + c > a \)

In this problem, the sides of the triangle are given as:
[tex]\[ a = 2x + 2, \quad b = x + 3, \quad \text{and} \quad c = n \][/tex]

Applying the triangle inequality theorem, we get the following three inequalities:

1. \( (2x + 2) + (x + 3) > n \)
2. \( (2x + 2) + n > x + 3 \)
3. \( (x + 3) + n > 2x + 2 \)

We will solve each inequality step-by-step:

### Inequality 1: \( (2x + 2) + (x + 3) > n \)
[tex]\[ 2x + 2 + x + 3 > n \][/tex]
[tex]\[ 3x + 5 > n \][/tex]
or equivalently,
[tex]\[ n < 3x + 5 \][/tex]

### Inequality 2: \( (2x + 2) + n > x + 3 \)
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
[tex]\[ 2x + 2 + n - x > 3 \][/tex]
[tex]\[ x + 2 + n > 3 \][/tex]
[tex]\[ n > 3 - x - 2 \][/tex]
[tex]\[ n > x - 1 \][/tex]

### Inequality 3: \( (x + 3) + n > 2x + 2 \)
[tex]\[ x + 3 + n > 2x + 2 \][/tex]
[tex]\[ n > 2x + 2 - x - 3 \][/tex]
[tex]\[ n > x - 1 \][/tex]

Even though we derived the same inequality for the second part, it reconfirms our previous finding.
Thus, combining both inequalities from above, we get:

[tex]\[ x - 1 < n < 3x + 5 \][/tex]

This means that \( n \) must be greater than \( x - 1 \) and less than \( 3x + 5 \) to satisfy all three conditions of the triangle inequality.

The correct solution is:

[tex]\[ x - 1 < n < 3x + 5 \][/tex]

Thus, the correct answer is:
[tex]\[ \boxed{x - 1 < n < 3x + 5} \][/tex]

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