1. **State the problem:** Solve the equation $$\frac{6}{5}(2x + 1) - \frac{7}{10}(x - 7) = 1$$ for $x$. 2. **Write the equation clearly:** $$\frac{6}{5}(2x + 1) - \frac{7}{10}(x - 7) = 1$$ 3. **Distribute the fractions:** $$\frac{6}{5} \times 2x + \frac{6}{5} \times 1 - \frac{7}{10} \times x + \frac{7}{10} \times 7 = 1$$ 4. **Calculate each term:** $$\frac{12x}{5} + \frac{6}{5} - \frac{7x}{10} + \frac{49}{10} = 1$$ 5. **Find a common denominator to combine like terms:** The common denominator for 5 and 10 is 10. Rewrite terms: $$\frac{24x}{10} + \frac{12}{10} - \frac{7x}{10} + \frac{49}{10} = 1$$ 6. **Combine like terms:** $$\left(\frac{24x}{10} - \frac{7x}{10}\right) + \left(\frac{12}{10} + \frac{49}{10}\right) = 1$$ $$\frac{17x}{10} + \frac{61}{10} = 1$$ 7. **Isolate $x$ by subtracting $\frac{61}{10}$ from both sides:** $$\frac{17x}{10} + \cancel{\frac{61}{10}} - \cancel{\frac{61}{10}} = 1 - \frac{61}{10}$$ $$\frac{17x}{10} = \frac{10}{10} - \frac{61}{10} = -\frac{51}{10}$$ 8. **Solve for $x$ by multiplying both sides by the reciprocal of $\frac{17}{10}$, which is $\frac{10}{17}$:** $$x = -\frac{51}{10} \times \frac{10}{17}$$ $$x = -\frac{51 \cancel{10}}{\cancel{10} 17} = -\frac{51}{17}$$ 9. **Simplify the fraction:** $$-\frac{51}{17} = -3$$ **Final answer:** $$x = -3$$