We are given the equation $|3x - 4| - 2 = 3$ and asked to solve for $x$.

1. Problem Description

We are given the equation $|3x - 4| - 2 = 3$ and asked to solve for $x$.

2. Solution Steps

First, we isolate the absolute value term by adding $2$ to both sides of the equation:

$|3x - 4| - 2 + 2 = 3 + 2$

$|3x - 4| = 5$

Now, we consider the two possible cases for the expression inside the absolute value:

Case 1: $3x - 4 = 5$

Add $4$ to both sides:

$3x - 4 + 4 = 5 + 4$

$3x = 9$

Divide both sides by $3$:

$x = \frac{9}{3}$

$x = 3$

Case 2: $3x - 4 = -5$

Add $4$ to both sides:

$3x - 4 + 4 = -5 + 4$

$3x = -1$

Divide both sides by $3$:

$x = \frac{-1}{3}$

$x = -\frac{1}{3}$

3. Final Answer

The solutions are $x = 3$ and $x = -\frac{1}{3}$.