1. **State the problem:** We need to solve the equation $$30x - \left(5a_5 + 8a_4 + 6a_3 + \frac{8}{3}a_2 + \frac{2}{3}a_1\right) + \left(a_4 - \frac{1}{2}a_2 + \frac{1}{24}a_0\right) = 0$$ for the given values: $$a_0 = -1, a_1 = 1, a_2 = 1, a_3 = \frac{1}{2}, a_4 = \frac{1}{2}, a_5 = \frac{31}{60}$$ 2. **Substitute the given values:** Calculate each term inside the parentheses: $$5a_5 = 5 \times \frac{31}{60} = \frac{155}{60}$$ $$8a_4 = 8 \times \frac{1}{2} = 4$$ $$6a_3 = 6 \times \frac{1}{2} = 3$$ $$\frac{8}{3}a_2 = \frac{8}{3} \times 1 = \frac{8}{3}$$ $$\frac{2}{3}a_1 = \frac{2}{3} \times 1 = \frac{2}{3}$$ Sum these: $$5a_5 + 8a_4 + 6a_3 + \frac{8}{3}a_2 + \frac{2}{3}a_1 = \frac{155}{60} + 4 + 3 + \frac{8}{3} + \frac{2}{3}$$ 3. **Find common denominator and sum:** Convert all to fractions with denominator 60: $$\frac{155}{60} + \frac{240}{60} + \frac{180}{60} + \frac{160}{60} + \frac{40}{60} = \frac{155 + 240 + 180 + 160 + 40}{60} = \frac{775}{60}$$ 4. **Calculate the second parentheses:** $$a_4 - \frac{1}{2}a_2 + \frac{1}{24}a_0 = \frac{1}{2} - \frac{1}{2} \times 1 + \frac{1}{24} \times (-1) = \frac{1}{2} - \frac{1}{2} - \frac{1}{24} = -\frac{1}{24}$$ 5. **Rewrite the equation:** $$30x - \frac{775}{60} - \frac{1}{24} = 0$$ 6. **Combine constants:** Find common denominator 120: $$-\frac{775}{60} - \frac{1}{24} = -\frac{1550}{120} - \frac{5}{120} = -\frac{1555}{120}$$ 7. **Equation becomes:** $$30x - \frac{1555}{120} = 0$$ 8. **Isolate $x$:** $$30x = \frac{1555}{120}$$ Divide both sides by 30: $$x = \frac{1555}{120 \times 30} = \frac{1555}{3600}$$ Simplify numerator and denominator by 5: $$x = \frac{\cancel{1555}^{311}}{\cancel{3600}^{720}}$$ 9. **Final answer:** $$x = \frac{311}{720}$$