To find the value for which the expression \( \frac{4 + 5x}{9x - 6} \) is undefined, we need to determine when the denominator \( 9x - 6 \) equals zero.
Set the denominator equal to zero and solve for \( x \):
\[ 9x - 6 = 0 \]
Add 6 to both sides:
\[ 9x = 6 \]
Now, divide both sides by 9:
\[ x = \frac{6}{9} \]
This simplifies to:
\[ x = \frac{2}{3} \]
Thus, the undefined value is:
\[ \boxed{\frac{2}{3}} \]