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 triangle inequality theorem states that for a triangle with side lengths \(a\), \(b\), and \(c\), the following must hold: \(|a - b| < c < a + b\). Here, \(a = 2x + 2\), \(b = x + 3\), and \(c = n\).

Step2: Find the difference of the two sides

First, calculate \(|(2x + 2)-(x + 3)|\). Simplify the expression inside the absolute value: \((2x + 2)-(x + 3)=2x + 2 - x - 3=x - 1\). Since side lengths are positive, we can consider the positive difference (as length is positive, the inequality will hold with the positive difference for the lower bound).

Step3: Find the sum of the two sides

Calculate the sum of the two sides: \((2x + 2)+(x + 3)=2x + 2 + x + 3 = 3x + 5\).

Step4: Apply the triangle inequality

Using the triangle inequality theorem, we have \(|(2x + 2)-(x + 3)| < n < (2x + 2)+(x + 3)\), which simplifies to \(x - 1 < n < 3x + 5\). Also, we need to ensure that the side lengths are positive, but since we are looking for the range of \(n\) in terms of \(x\) and the other sides are expressed in terms of \(x\) (implying \(x\) is such that the sides are positive), this inequality holds.

Answer:

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