To solve the equations involving the expression \( |3x + 1/6| \), let's break them down step by step.

### First Equation:
\[ 5 - |3x + \frac{1}{6}| = 1 \]

1. Subtract 5 from both sides:
\[
-|3x + \frac{1}{6}| = 1 - 5
\]
\[
-|3x + \frac{1}{6}| = -4
\]

2. Multiply both sides by -1:
\[
|3x + \frac{1}{6}| = 4
\]

3. Now, we can split this absolute value equation into two cases:
- Case 1:
\[
3x + \frac{1}{6} = 4
\]
- Case 2:
\[
3x + \frac{1}{6} = -4
\]

**For Case 1:**
\[
3x = 4 - \frac{1}{6}
\]
\[
3x = \frac{24}{6} - \frac{1}{6} = \frac{23}{6}
\]
\[
x = \frac{23}{18}
\]

**For Case 2:**
\[
3x = -4 - \frac{1}{6}
\]
\[
3x = -\frac{24}{6} - \frac{1}{6} = -\frac{25}{6}
\]
\[
x = -\frac{25}{18}
\]

### So the solutions for the first equation are:
- \( x = \frac{23}{18} \)
- \( x = -\frac{25}{18} \)

### Second Equation:
\[ \frac{25 - |3x + \frac{1}{6}|}{1} = \frac{3}{2} \]

1. Rearrange the equation:
\[
25 - |3x + \frac{1}{6}| = \frac{3}{2}
\]

2. Subtract 25 from both sides:
\[
-|3x + \frac{1}{6}| = \frac{3}{2} - 25
\]
\[
-|3x + \frac{1}{6}| = \frac{3}{2} - \frac{50}{2}
\]
\[
-|3x + \frac{1}{6}| = -\frac{47}{2}
\]

3. Multiply both sides by -1:
\[
|3x + \frac{1}{6}| = \frac{47}{2}
\]

4. Again, we can break this into two cases:
- Case 1:
\[
3x + \frac{1}{6} = \frac{47}{2}
\]
- Case 2:
\[
3x + \frac{1}{6} = -\frac{47}{2}
\]

**For Case 1:**
\[
3x = \frac{47}{2} - \frac{1}{6}
\]
First, we need a common denominator. The common denominator of 2 and 6 is 6:
\[
\frac{47}{2} = \frac{141}{6}
\]
\[
3x = \frac{141}{6} - \frac{1}{6} = \frac{140}{6} = \frac{70}{3}
\]
\[
x = \frac{70}{9}
\]

**For Case 2:**
\[
3x = -\frac{47}{2} - \frac{1}{6}
\]
Again, use a common denominator of 6:
\[
-\frac{47}{2} = -\frac{141}{6}
\]
\[
3x = -\frac{141}{6} - \frac{1}{6} = -\frac{142}{6} = -\frac{71}{3}
\]
\[
x = -\frac{71}{9}
\]

### So the solutions for the second equation are:
- \( x = \frac{70}{9} \)
- \( x = -\frac{71}{9} \)

### Summary of Solutions:
1. From the first equation:
- \( x = \frac{23}{18} \)
- \( x = -\frac{25}{18} \)

2. From the second equation:
- \( x = \frac{70}{9} \)
- \( x = -\frac{71}{9} \)