Solve the following inequations:
(i) `|3x-2|le(1)/(2)`
(ii) `|x+(1)/(4)|gt(7)/(4)`
(iii) `|(3x-4)/(2)|le(5)/(12)`
(iv) `|4-x|+1lt3`.

Answer

Let's solve each of the given inequations step by step.

### (i) Solve `|3x - 2| ≤ (1/2)`

1. **Set up the inequality**:
\[ -\frac{1}{2} \leq 3x - 2 \leq \frac{1}{2} \]

2. **Break it into two parts**:
- From the left side:
\[ 3x - 2 \geq -\frac{1}{2} \]
- From the right side:
\[ 3x - 2 \leq \frac{1}{2} \]

3. **Solve the left side**:
\[ 3x \geq -\frac{1}{2} + 2 \implies 3x \geq \frac{3}{2} \implies x \geq \frac{1}{2} \]

4. **Solve the right side**:
\[ 3x \leq \frac{1}{2} + 2 \implies 3x \leq \frac{5}{2} \implies x \leq \frac{5}{6} \]

5. **Combine the results**:
\[ \frac{1}{2} \leq x \leq \frac{5}{6} \]

**Final Solution**:
\[ x \in \left[\frac{1}{2}, \frac{5}{6}\right] \]