To solve for \( x \) in the given equation:

\[
\frac{5 + x}{3x - 6} - \frac{2x - 3}{2x - 4} = \frac{1}{2}
\]

First, find a common denominator for the fractions on the left side. The common denominator for \( 3x - 6 \) and \( 2x - 4 \) is \( (3x - 6)(2x - 4) \).

Rewrite each fraction with the common denominator:

\[
\frac{(5 + x)(2x - 4)}{(3x - 6)(2x - 4)} - \frac{(2x - 3)(3x - 6)}{(3x - 6)(2x - 4)} = \frac{1}{2}
\]

Combine the fractions:

\[
\frac{(5 + x)(2x - 4) - (2x - 3)(3x - 6)}{(3x - 6)(2x - 4)} = \frac{1}{2}
\]

Simplify the numerators:

\[
(5 + x)(2x - 4) = 10x - 20 + 2x^2 - 4x = 2x^2 + 6x - 20
\]

\[
(2x - 3)(3x - 6) = 6x^2 - 12x - 9x + 18 = 6x^2 - 21x + 18
\]

Subtract the second numerator from the first:

\[
2x^2 + 6x - 20 - (6x^2 - 21x + 18) = 2x^2 + 6x - 20 - 6x^2 + 21x - 18 = -4x^2 + 27x - 38
\]

Now the equation is:

\[
\frac{-4x^2 + 27x - 38}{(3x - 6)(2x - 4)} = \frac{1}{2}
\]

Cross-multiply to solve for \( x \):

\[
2(-4x^2 + 27x - 38) = (3x - 6)(2x - 4)
\]

\[
-8x^2 + 54x - 76 = 6x^2 - 24x - 12x + 24
\]

\[
-8x^2 + 54x - 76 = 6x^2 - 36x + 24
\]

Combine like terms:

\[
-8x^2 + 54x - 76 - 6x^2 + 36x - 24 = 0
\]

\[
-14x^2 + 90x - 100 = 0
\]

Divide the entire equation by -2 to simplify:

\[
7x^2 - 45x + 50 = 0
\]

Solve the quadratic equation using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 7 \), \( b = -45 \), and \( c = 50 \):

\[
x = \frac{45 \pm \sqrt{(-45)^2 - 4 \cdot 7 \cdot 50}}{2 \cdot 7}
\]

\[
x = \frac{45 \pm \sqrt{2025 - 1400}}{14}
\]

\[
x = \frac{45 \pm \sqrt{625}}{14}
\]

\[
x = \frac{45 \pm 25}{14}
\]

This gives us two solutions:

\[
x = \frac{70}{14} = 5
\]

\[
x = \frac{20}{14} = \frac{10}{7}
\]

So, the solutions are \( x = 5 \) and \( x = \frac{10}{7} \).