1. **State the problem:** Solve the inequality $$ (7x - 2)^2 - 7(x - \frac{2}{7})(7x + 1) + \frac{x}{2} > 3(7x - 2) + \frac{1}{7} $$ with the condition $$ x < \frac{2}{7} $$. 2. **Expand and simplify each term:** - Expand $$ (7x - 2)^2 = 49x^2 - 28x + 4 $$ - Expand $$ -7(x - \frac{2}{7})(7x + 1) $$: First, expand inside the parentheses: $$ (x - \frac{2}{7})(7x + 1) = 7x^2 + x - 2x - \frac{2}{7} = 7x^2 - x - \frac{2}{7} $$ Multiply by $$ -7 $$: $$ -7(7x^2 - x - \frac{2}{7}) = -49x^2 + 7x + 2 $$ - The term $$ \frac{x}{2} $$ remains as is. - The right side is $$ 3(7x - 2) + \frac{1}{7} = 21x - 6 + \frac{1}{7} = 21x - \frac{41}{7} $$ 3. **Rewrite the inequality:** $$ 49x^2 - 28x + 4 - 49x^2 + 7x + 2 + \frac{x}{2} > 21x - \frac{41}{7} $$ 4. **Combine like terms on the left:** $$ (49x^2 - 49x^2) + (-28x + 7x + \frac{x}{2}) + (4 + 2) > 21x - \frac{41}{7} $$ $$ 0 + (-21x + \frac{x}{2}) + 6 > 21x - \frac{41}{7} $$ 5. **Simplify the x terms:** $$ -21x + \frac{x}{2} = -\frac{42x}{2} + \frac{x}{2} = -\frac{41x}{2} $$ So inequality becomes: $$ -\frac{41x}{2} + 6 > 21x - \frac{41}{7} $$ 6. **Bring all terms to one side:** $$ -\frac{41x}{2} + 6 - 21x + \frac{41}{7} > 0 $$ 7. **Combine x terms:** $$ -\frac{41x}{2} - 21x = -\frac{41x}{2} - \frac{42x}{2} = -\frac{83x}{2} $$ 8. **Combine constants:** $$ 6 + \frac{41}{7} = \frac{42}{7} + \frac{41}{7} = \frac{83}{7} $$ 9. **Inequality is now:** $$ -\frac{83x}{2} + \frac{83}{7} > 0 $$ 10. **Multiply both sides by 14 (LCM of 2 and 7) to clear denominators:** $$ 14 \times \left(-\frac{83x}{2} + \frac{83}{7}\right) > 14 \times 0 $$ $$ -7 \times 83x + 2 \times 83 > 0 $$ $$ -581x + 166 > 0 $$ 11. **Isolate x:** $$ -581x > -166 $$ 12. **Divide both sides by -581, reversing inequality sign because divisor is negative:** $$ x < \frac{166}{581} $$ 13. **Simplify fraction if possible:** 166 and 581 share a factor of 1 only, so fraction stays as is. 14. **Recall the domain restriction:** $$ x < \frac{2}{7} \approx 0.2857 $$ 15. **Compare $$ \frac{166}{581} \approx 0.2857 $$ to $$ \frac{2}{7} \approx 0.2857 $$:** They are approximately equal, but $$ \frac{166}{581} \approx 0.2857 $$ is slightly less than $$ \frac{2}{7} $$ (since 2/7 = 0.285714...). 16. **Final solution considering domain:** $$ x < \frac{166}{581} $$ **Answer:** $$ \boxed{x < \frac{166}{581}} $$