Simplify the left side of the inequality.

$$3x - \frac{1}{4} \left(\frac{1}{2}\right) > \frac{1}{4} - x$$

Multiply the fractions on the left side:

$$3x - \frac{1}{8} > \frac{1}{4} - x$$

Add $x$ to both sides of the inequality:

$$3x + x - \frac{1}{8} > \frac{1}{4}$$ $$4x - \frac{1}{8} > \frac{1}{4}$$

Add $\frac{1}{8}$ to both sides of the inequality:

$$4x > \frac{1}{4} + \frac{1}{8}$$

Find a common denominator for $\frac{1}{4}$ and $\frac{1}{8}$, which is 8.

$$\frac{1}{4} = \frac{1 \times 2}{4 \times 2} = \frac{2}{8}$$

Now add the fractions:

$$4x > \frac{2}{8} + \frac{1}{8}$$ $$4x > \frac{3}{8}$$

Divide both sides by 4:

$$x > \frac{3}{8 \times 4}$$ $$x > \frac{3}{32}$$

The inequality $x > \frac{3}{32}$ means that $x$ can be any value strictly greater than $\frac{3}{32}$. In interval notation, this is written as:

$$\left(\frac{3}{32}, \infty\right)$$