Question Point R is on line segment \( \overline{Q S} \). Given \( Q S=2 x+2, Q R=3 x-5 \), and \( R S=5 x-5 \), determine the numerical length of \( \overline{Q R} \).

To find the numerical length of \( \overline{QR} \), we start by noting that since point \( R \) is on segment \( \overline{QS} \), we can express the relationship as: \[ QR + RS = QS \] Substituting the given expressions into that equation, we have: \[ (3x - 5) + (5x - 5) = 2x + 2 \] Simplifying the left side: \[ 3x - 5 + 5x - 5 = 8x - 10 \] Now, setting this equal to the right side: \[ 8x - 10 = 2x + 2 \] Next, we solve for \( x \): \[ 8x - 2x = 2 + 10 \] This simplifies to: \[ 6x = 12 \] So, \[ x = 2 \] Now that we have \( x \), we can find the length of \( \overline{QR} \): \[ QR = 3x - 5 = 3(2) - 5 = 6 - 5 = 1 \] Thus, the numerical length of \( \overline{QR} \) is \[ \boxed{1}. \]