1. **State the problem:** Simplify the expression $$\frac{1}{4} \times \left| \left( \frac{3}{2} + \frac{1}{4} \right) \div \sqrt{\frac{49}{4}} - 1 \right|.$$\n\n2. **Recall the rules:**\n- Addition of fractions requires a common denominator.\n- Division by a square root means dividing by the positive root.\n- Absolute value makes the expression inside non-negative.\n- Multiplication by a fraction scales the result.\n\n3. **Calculate inside the absolute value:**\nFirst, add the fractions inside the parentheses:\n$$\frac{3}{2} + \frac{1}{4} = \frac{6}{4} + \frac{1}{4} = \frac{7}{4}.$$\n\n4. **Calculate the square root:**\n$$\sqrt{\frac{49}{4}} = \frac{\sqrt{49}}{\sqrt{4}} = \frac{7}{2}.$$\n\n5. **Divide the sum by the square root:**\n$$\frac{7}{4} \div \frac{7}{2} = \frac{7}{4} \times \frac{2}{7} = \cancel{\frac{7}{4}} \times \cancel{\frac{2}{7}} = \frac{2}{4} = \frac{1}{2}.$$\n\n6. **Subtract 1:**\n$$\frac{1}{2} - 1 = \frac{1}{2} - \frac{2}{2} = -\frac{1}{2}.$$\n\n7. **Apply absolute value:**\n$$\left| -\frac{1}{2} \right| = \frac{1}{2}.$$\n\n8. **Multiply by $\frac{1}{4}$:**\n$$\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}.$$\n\n**Final answer:** $$\frac{1}{8}.$$