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. x - 1 < n < 3x + 5 n = 3x + 5 n = x - 1 3x + 5 < n < x - 1

Explanation:

Step1: Apply triangle - inequality theorem (sum of two sides > third side)

Let the sides of the triangle be \(a = 2x + 2\), \(b=x + 3\), and \(c=n\).
First, \(a + b>c\), so \((2x + 2)+(x + 3)>n\), which simplifies to \(3x+5>n\).

Step2: Apply another triangle - inequality

\(a + c>b\), so \((2x + 2)+n>x + 3\), which simplifies to \(n>x - 1\).

Step3: Apply the third triangle - inequality

\(b + c>a\), so \((x + 3)+n>2x + 2\), which also gives \(n>x - 1\) after simplification.

Answer:

A. \(x - 1 < n < 3x+5\)