1. **State the problem:** Simplify the expression $\left(\frac{5}{9}x + \frac{1}{8}\right) + \left(\frac{4}{7}x - \frac{1}{8}\right)$. 2. **Write the expression:** $$\frac{5}{9}x + \frac{1}{8} + \frac{4}{7}x - \frac{1}{8}$$ 3. **Combine like terms:** Group the $x$ terms and the constants separately: $$\left(\frac{5}{9}x + \frac{4}{7}x\right) + \left(\frac{1}{8} - \frac{1}{8}\right)$$ 4. **Simplify the constants:** $$\frac{1}{8} - \frac{1}{8} = 0$$ 5. **Add the $x$ terms:** Find a common denominator for $\frac{5}{9}$ and $\frac{4}{7}$. The least common denominator is $63$. $$\frac{5}{9} = \frac{5 \times 7}{9 \times 7} = \frac{35}{63}$$ $$\frac{4}{7} = \frac{4 \times 9}{7 \times 9} = \frac{36}{63}$$ 6. **Add the fractions:** $$\frac{35}{63}x + \frac{36}{63}x = \frac{35 + 36}{63}x = \frac{71}{63}x$$ 7. **Final simplified expression:** $$\frac{71}{63}x + 0 = \frac{71}{63}x$$ **Answer:** C. $\frac{71}{63}x$