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A triangle has side lengths measuring [tex]2x + 2 \text{ ft}[/tex], [tex]x + 3 \text{ ft}[/tex], and [tex]n \text{ ft}[/tex]. 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 :

GinnyAnswer
To find the possible values for [tex]\( n \)[/tex] in a triangle with side lengths [tex]\( 2x + 2 \, \text{ft} \)[/tex], [tex]\( x + 3 \, \text{ft} \)[/tex], and [tex]\( n \, \text{ft} \)[/tex], 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.

We have three inequalities to consider:
1. [tex]\( (2x + 2) + (x + 3) > n \)[/tex]
2. [tex]\( (2x + 2) + n > (x + 3) \)[/tex]
3. [tex]\( (x + 3) + n > (2x + 2) \)[/tex]

Let’s solve each inequality step-by-step:

1. For the first inequality:
[tex]\[ (2x + 2) + (x + 3) > n \][/tex]
[tex]\[ 3x + 5 > n \][/tex]
[tex]\[ n < 3x + 5 \][/tex]

2. For the second inequality:
[tex]\[ (2x + 2) + n > (x + 3) \][/tex]
[tex]\[ 2x + 2 + n > x + 3 \][/tex]
[tex]\[ 2x + n + 2 > x + 3 \][/tex]
[tex]\[ 2x + n > x + 1 \][/tex]
[tex]\[ n > x - 1 \][/tex]

3. For the third inequality:
[tex]\[ (x + 3) + n > (2x + 2) \][/tex]
[tex]\[ x + n + 3 > 2x + 2 \][/tex]
[tex]\[ n + x + 3 > 2x + 2 \][/tex]
[tex]\[ n + 3 > x + 2 \][/tex]
[tex]\[ n > x - 1 \][/tex]

Notice that the result from the third inequality [tex]\( n > x - 1 \)[/tex] is the same as the result from the second inequality. Therefore, the combined simplified inequalities are:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]

Thus, we have concluded that the expression representing the possible values of [tex]\( n \)[/tex] is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]

Therefore, the correct answer is:
[tex]\[ x-1

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