1. The problem is to find the value of $x$ if $2|2x-1|=\frac{1}{4}$. 2. First, divide both sides of the equation by 2 to isolate the absolute value: $$|2x-1|=\frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$$ 3. The definition of absolute value tells us that if $|A|=B$, then $A=B$ or $A=-B$. So, $$2x-1=\frac{1}{8} \quad \text{or} \quad 2x-1=-\frac{1}{8}$$ 4. Solve each equation for $x$: \begin{align*} 2x-1 &= \frac{1}{8} \\ 2x &= \frac{1}{8} + 1 = \frac{1}{8} + \frac{8}{8} = \frac{9}{8} \\ x &= \frac{9}{8} \times \frac{1}{2} = \frac{9}{16} \end{align*} \begin{align*} 2x-1 &= -\frac{1}{8} \\ 2x &= -\frac{1}{8} + 1 = -\frac{1}{8} + \frac{8}{8} = \frac{7}{8} \\ x &= \frac{7}{8} \times \frac{1}{2} = \frac{7}{16} \end{align*} 5. The solutions are $x = \frac{9}{16}$ or $x = \frac{7}{16}$.