The equation provided in the image is:

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

To solve this absolute value equation, we need to consider all possible cases. Absolute value equations are solved by analyzing the conditions under which the expressions inside the absolute value are positive or negative.

Case 1: $3x - 4 = 2 - 3x$

Simplify: $3x + 3x = 2 + 4 \quad \Rightarrow \quad 6x = 6 \quad \Rightarrow \quad x = 1$

Case 2: $3x - 4 = -(2 - 3x)$

Simplify: $3x - 4 = -2 + 3x \quad \Rightarrow \quad 3x - 3x = -2 + 4 \quad \Rightarrow \quad 0x = 2$

This case is not valid since $0x = 2$ is a contradiction.

Case 3: $-(3x - 4) = 2 - 3x$

Simplify: $-3x + 4 = 2 - 3x \quad \Rightarrow \quad 4 = 2$

This case is also invalid due to a contradiction.

Case 4: $-(3x - 4) = -(2 - 3x)$

Simplify: $-3x + 4 = -2 + 3x \quad \Rightarrow \quad -3x - 3x = -2 - 4 \quad \Rightarrow \quad -6x = -6 \quad \Rightarrow \quad x = 1$

Final Solution:

The only solution to the equation is: $\boxed{x = 1}$

Would you like additional explanation or verification? Here are some related questions to deepen understanding:

1. How do we determine the number of cases for absolute value equations?
2. What happens when the absolute values contain more than two variables?
3. Why do some cases lead to contradictions in absolute value problems?
4. Can graphing the functions help verify the solution of the equation?
5. How are absolute value inequalities different from absolute value equations?

Tip: Always check each case carefully for contradictions or invalid solutions when solving absolute value equations!