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 any triangle with side lengths \( a \), \( b \), and \( c \), the following must hold: \( |a - b| < c < a + b \). Here, the sides are \( a = 2x + 2 \), \( b = x + 3 \), 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 lengths are positive, and we assume \( x \) is such that \( x-1>0 \) (otherwise the triangle inequality would not hold in a meaningful way for positive length \( n \)), we have \( |(2x + 2)-(x + 3)|=x - 1 \).

Step3: Find the sum of the two sides

Next, calculate \( (2x + 2)+(x + 3) \). Combine like terms: \( 2x+2+x + 3=3x + 5 \).

Step4: Apply Triangle Inequality

By the Triangle Inequality Theorem, \( |a - b| < n < a + b \), so substituting the values we found, we get \( x - 1 < n < 3x + 5 \).

Answer:

\( x - 1 < n < 3x + 5 \) (corresponding to the first option)