8) Find \( x \) if \( B C=x+20, A B=x+17 \) and \( A C=21 \)

To solve for \( x \), we can apply the triangle inequality, which states that the sum of the lengths of any two sides of a triangle must be greater than the third side. In this case, let’s consider triangle \( ABC \) with sides \( AB = x + 17 \), \( BC = x + 20 \), and \( AC = 21 \). By applying the triangle inequality: 1. \( AB + BC > AC \): \[ (x + 17) + (x + 20) > 21 \] Simplifying this gives: \[ 2x + 37 > 21 \] \[ 2x > -16 \] \[ x > -8 \] 2. \( AB + AC > BC \): \[ (x + 17) + 21 > (x + 20) \] Simplifying this gives: \[ x + 38 > x + 20 \] This simplifies to: \[ 38 > 20 \] (This is always true). 3. \( AC + BC > AB \): \[ 21 + (x + 20) > (x + 17) \] Simplifying this gives: \[ 41 + x > x + 17 \] Which simplifies to: \[ 41 > 17 \] (This is also always true). Since the inequalities involving \( x \) do not conflict, the only restriction we have is \( x > -8 \). Thus, the integrated information shows how the properties of triangles help in determining valid ranges for side lengths.