1. We are given two expressions to simplify and evaluate. 2. For the first expression, $x = \frac{1}{4} + \frac{1}{8} + \frac{3}{2}$. 3. To add these fractions, find a common denominator. The denominators are 4, 8, and 2. The least common denominator is 8. 4. Convert each fraction: - $\frac{1}{4} = \frac{2}{8}$ - $\frac{1}{8} = \frac{1}{8}$ - $\frac{3}{2} = \frac{12}{8}$ 5. Add the fractions: $$x = \frac{2}{8} + \frac{1}{8} + \frac{12}{8} = \frac{15}{8}$$ 6. So, the simplified value is $x = \frac{15}{8}$. 7. For the second expression, $x = 5 \sin \frac{\pi}{6} - 3 \sqrt{2} \sin \frac{\pi}{4} + \frac{1}{4} \tan \frac{\pi}{4}$. 8. Recall the exact values: - $\sin \frac{\pi}{6} = \frac{1}{2}$ - $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ - $\tan \frac{\pi}{4} = 1$ 9. Substitute these values: $$x = 5 \times \frac{1}{2} - 3 \sqrt{2} \times \frac{\sqrt{2}}{2} + \frac{1}{4} \times 1$$ 10. Simplify each term: - $5 \times \frac{1}{2} = \frac{5}{2}$ - $3 \sqrt{2} \times \frac{\sqrt{2}}{2} = 3 \times \frac{2}{2} = 3$ - $\frac{1}{4} \times 1 = \frac{1}{4}$ 11. Combine the terms: $$x = \frac{5}{2} - 3 + \frac{1}{4}$$ 12. Convert to a common denominator 4: - $\frac{5}{2} = \frac{10}{4}$ - $3 = \frac{12}{4}$ - $\frac{1}{4} = \frac{1}{4}$ 13. Add and subtract: $$x = \frac{10}{4} - \frac{12}{4} + \frac{1}{4} = \frac{10 - 12 + 1}{4} = \frac{-1}{4}$$ 14. Final simplified value is $x = -\frac{1}{4}$. Therefore, the answers are: - a) $x = \frac{15}{8}$ - b) $x = -\frac{1}{4}$