1. **Problem statement:** Given the functions $f(x) = 5x + 1$ and $g(x) = \frac{x - 4}{2}$, find the value of $x$ if $fg(x) = 11$. 2. **Recall the composition of functions:** $$fg(x) = f(g(x))$$ This means we first apply $g$ to $x$, then apply $f$ to the result. 3. **Calculate $g(x)$:** $$g(x) = \frac{x - 4}{2}$$ 4. **Apply $f$ to $g(x)$:** $$fg(x) = f\left(\frac{x - 4}{2}\right) = 5 \times \frac{x - 4}{2} + 1$$ 5. **Set $fg(x) = 11$ and solve for $x$:** $$5 \times \frac{x - 4}{2} + 1 = 11$$ 6. **Subtract 1 from both sides:** $$5 \times \frac{x - 4}{2} = 11 - 1$$ $$5 \times \frac{x - 4}{2} = 10$$ 7. **Multiply both sides by 2 to clear the denominator:** $$2 \times 5 \times \frac{x - 4}{2} = 2 \times 10$$ $$\cancel{2} \times 5 \times \frac{x - 4}{\cancel{2}} = 20$$ $$5(x - 4) = 20$$ 8. **Divide both sides by 5:** $$\frac{5(x - 4)}{5} = \frac{20}{5}$$ $$\cancel{5}(x - 4)/\cancel{5} = 4$$ $$x - 4 = 4$$ 9. **Add 4 to both sides:** $$x = 4 + 4$$ $$x = 8$$ **Final answer:** $x = 8$