To solve the equation \( \log_9(2x - 1) = -1 \), we start by rewriting the logarithmic equation in its exponential form:
\[ 2x - 1 = 9^{-1} \]
Since \( 9^{-1} = \frac{1}{9} \), we can substitute this value into the equation:
\[ 2x - 1 = \frac{1}{9} \]
Next, we will solve for \( x \) by first adding 1 to both sides of the equation:
\[ 2x = \frac{1}{9} + 1 \]
To combine the fractions, we convert 1 into a fraction with a denominator of 9:
\[ 1 = \frac{9}{9} \]
Now, we have:
\[ 2x = \frac{1}{9} + \frac{9}{9} = \frac{10}{9} \]
Next, we divide both sides by 2:
\[ x = \frac{10}{9} \times \frac{1}{2} = \frac{10}{18} = \frac{5}{9} \]
So, the solution to the equation is:
\[ x = \frac{5}{9} \]
Thus, the response is:
\[ x = \frac{5}{9} \]
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