First, we move all the terms to the left-hand side to get
\[3x^2 - 11x -7 = 0.\]We can easily verify that 1 is not a root, so that
\[\frac{1}{a - 1} + \frac{1}{b - 1} = \frac{a + b - 2}{ab - (a + b) + 1}.\]By Vieta's formulas, $a + b = \frac{11}{3}$ and $ab = -\frac{7}{3}.$ Hence,
\[\frac{11/3 - 2}{-7/3 - 11/3 + 1} = \frac{\frac{5}{3}}{\frac{-17}{3}} = \boxed{-\frac{5}{17}}.\]
Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a-1}+\frac{1}{b-1}.$
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