To find the value of \( f(5) \), we can use the given equation and the value of \( f(6) \):
\[ f(x) - f(x + 1) = 2x - 5 \]
Given that \( f(6) = 12 \), we can substitute \( x = 6 \) into the equation:
\[ f(6) - f(6 + 1) = 2(6) - 5 \]
\[ 12 - f(7) = 12 - 5 \]
\[ - f(7) = 7 \]
\[ f(7) = -7 \]
Now, to find \( f(5) \), we can use \( x = 6 \) and \( x = 5 \):
\[ f(6) - f(7) = 2(6) - 5 \]
\[ 12 - (-7) = 2(6) - 5 \]
\[ 19 = 12 - 5 \]
\[ f(5) = 19 \]
So, the value of \( f(5) \) is \( \boxed{19} \).
