1. **State the problem:** We need to find the length of segment $GF$ given points $G$, $F$, and $E$ on a horizontal line with $GF = (x + 16) + (x + 19)$ and the total length $GE = 19$. 2. **Set up the equation:** Since $G$, $F$, and $E$ are collinear and $F$ lies between $G$ and $E$, the sum of $GF$ and $FE$ equals $GE$. Here, $GF = x + 16$, $FE = x + 19$, and $GE = 19$. 3. **Write the equation:** $$GF + FE = GE$$ $$ (x + 16) + (x + 19) = 19 $$ 4. **Simplify the equation:** $$ x + 16 + x + 19 = 19 $$ $$ 2x + 35 = 19 $$ 5. **Solve for $x$:** $$ 2x + 35 = 19 $$ Subtract 35 from both sides: $$ 2x + \cancel{35} - \cancel{35} = 19 - 35 $$ $$ 2x = -16 $$ Divide both sides by 2: $$ \frac{2x}{\cancel{2}} = \frac{-16}{\cancel{2}} $$ $$ x = -8 $$ 6. **Find $GF$:** $$ GF = x + 16 = -8 + 16 = 8 $$ **Final answer:** $$ GF = 8 $$