Question

a triangle has side lengths measuring ( 2x + 2 ) ft, ( x + 3 ) ft, and ( n ) ft. which expression represents the possible values of ( n ), in feet? express your answer in simplest terms. ( n = x - 1 ) ( x - 1 lt n lt 3x + 5 ) ( 3x + 5 lt n lt x - 1 ) ( n = 3x + 5 )

Explanation:

Step1: Use triangle - inequality theorem

The triangle - inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), \(|a - b|\lt c\lt a + b\). Let \(a=(2x + 2)\), \(b=(x + 3)\).
First, find \(a + b\):

$$
LATEXBLOCK0
$$

Second, find \(|a - b|\):

$$
LATEXBLOCK1
$$

Since \(x+3\gt0\) and \(2x + 2\gt0\) (side - length is positive), when \(x\gt1\), \(|x - 1|=x - 1\)

Step2: Determine the range of \(n\)

By the triangle - inequality theorem \(|a - b|\lt n\lt a + b\), substituting \(a=(2x + 2)\), \(b=(x + 3)\)
We get \(x - 1\lt n\lt3x + 5\)

Answer:

\(x - 1\lt n\lt3x + 5\)