Step 1: Understanding the Concept:
Express all terms using base 3 and base 2 to group similar exponential terms.
Step 2: Key Formula or Approach:
\( a^m = a^n \Rightarrow m = n \). Rearrange bases to opposite sides.
Step 3: Detailed Explanation:
\( (3^2)^{x-1/2} - 2^{2x-2} = (2^2)^x - 3^{2x-3} \).
\( 3^{2x-1} - 2^{2x-2} = 2^{2x} - 3^{2x-3} \).
Rearranging:
\( 3^{2x-1} + 3^{2x-3} = 2^{2x} + 2^{2x-2} \).
Factor out the lowest powers:
\( 3^{2x-3}(3^2 + 1) = 2^{2x-2}(2^2 + 1) \).
\( 3^{2x-3}(10) = 2^{2x-2}(5) \).
\( 3^{2x-3} \cdot 2 = 2^{2x-2} \).
\( 3^{2x-3} = 2^{2x-2} / 2 = 2^{2x-3} \).
Since bases are different, the equality \( 3^A = 2^A \) only holds if \( A = 0 \).
\( 2x - 3 = 0 \Rightarrow x = 3/2 \).
Step 4: Final Answer:
The value of x is 3/2.