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: Recall triangle - inequality theorem

The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Let \(a = 2x + 2\), \(b=x + 3\), and \(c=n\).

1. \(a + b>c\): \((2x + 2)+(x + 3)>n\), which simplifies to \(3x+5>n\).
2. \(a + c>b\): \((2x + 2)+n>x + 3\), which simplifies to \(n>x - 1\).
3. \(b + c>a\): \((x + 3)+n>2x + 2\), which also simplifies to \(n>x - 1\) (after rearranging terms).

Answer:

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