Answered

A triangle has side lengths measuring [tex][tex]$2x + 2 \, \text{ft}$[/tex][/tex], [tex][tex]$x + 3 \, \text{ft}$[/tex][/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
Let's review the key property required for three lengths to form a triangle: the triangle inequality theorem. According to this theorem, for three sides of lengths [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] to form a triangle, the following inequalities must hold:

1. [tex]\(a + b > c\)[/tex]
2. [tex]\(a + c > b\)[/tex]
3. [tex]\(b + c > a\)[/tex]

In this question, we are given a triangle with side lengths [tex]\(2x + 2\)[/tex] feet, [tex]\(x + 3\)[/tex] feet, and [tex]\(n\)[/tex] feet. We need to determine the possible values of [tex]\(n\)[/tex] that satisfy the triangle inequality theorem.

Let's apply the inequalities one by one:

1. [tex]\( (2x + 2) + (x + 3) > n \)[/tex]
[tex]\[ 2x + 2 + x + 3 > n \\ 3x + 5 > n \\ n < 3x + 5 \][/tex]

2. [tex]\( (2x + 2) + n > (x + 3) \)[/tex]
[tex]\[ 2x + 2 + n > x + 3 \\ 2x + 2 + n - x > 3 \\ x + 2 + n > 3 \\ n > x + 1 - 2 \\ n > x + 1 \][/tex]

3. [tex]\( (x + 3) + n > (2x + 2) \)[/tex]
[tex]\[ x + 3 + n > 2x + 2 \\ n > 2x + 2 - x - 3 \\ n > x - 1 \][/tex]

By combining these results, we get the range:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]

Therefore, the expression that represents the possible values of [tex]\(n\)[/tex] is:
[tex]\[ x - 1 < n < 3x + 5 \][/tex]

So the correct answer is:

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

Other Questions