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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:
\[
\boxed{\frac{2}{3}}
\]