The two valid solutions are $ x = -\frac{10}{3} $ and $ x = \frac{14}{3} $. The largest is $ \frac{14}{3} $.

["The Two Valid Solutions Are: ( x = -\frac{10}{3} ) and ( x = \frac{14}{3} ) — The Largest Value Is ( \frac{14}{3} )", "When solving equations—especially linear or quadratic equations—identifying the correct and valid solutions is crucial to understanding the problem fully. In this context, two solutions have been derived:\n[\nx = -\frac{10}{3} \quad \ ext{and} \quad x = \frac{14}{3}\n]\nAmong these, the largest solution is clearly ( x = \frac{14}{3} ), which is significantly greater than ( -\frac{10}{3} ) on the real number line.", "### Why These Are Valid Solutions", "These values emerge as valid answers after performing proper algebraic operations—whether through factoring, applying the quadratic formula, or simplifying inequalities. Each satisfies the original equation(s), ensuring no extraneous or incorrect values were introduced during the solving process.", "### Comparing the Two Values", "To emphasize clarity:\n- ( \frac{14}{3} \approx 4.67 )\n- ( -\frac{10}{3} \approx -3.33 )", "Visualizing on a number line confirms that ( \frac{14}{3} ) lies far to the right of ( -\frac{10}{3} ), making it unequivocally the largest.", "### Practical Implications", "In real-world applications—such as optimization, physics, or economics—choosing the correct value ensures accurate modeling and decision-making. Whether modeling cost, profit, distance, or time, always verify which solution aligns best with context and constraints.", "### Conclusion", "With the two valid solutions established, confirming\n[\n\boxed{x = \frac{14}{3}} \quad \ ext{as the largest value}\n]\ncompletes the mathematical resolution and supports precise problem-solving moving forward.", "---\nKeywords: valid solutions, linear equation, quadratic equation solutions, largest solution, algebraic verification, real number comparison."]