To solve the equation \( \log_9(2x - 1) = -1 \), we first rewrite it in exponential form.
The equation \( \log_9(2x - 1) = -1 \) means:
\[ 2x - 1 = 9^{-1} \]
Calculating \( 9^{-1} \):
\[ 9^{-1} = \frac{1}{9} \]
So we can rewrite the equation as:
\[ 2x - 1 = \frac{1}{9} \]
Next, we solve for \( x \) by first adding 1 to both sides:
\[ 2x = \frac{1}{9} + 1 \]
We need a common denominator to add these fractions. The number 1 can be expressed as \( \frac{9}{9} \):
\[ 2x = \frac{1}{9} + \frac{9}{9} = \frac{1 + 9}{9} = \frac{10}{9} \]
Now, divide both sides by 2:
\[ x = \frac{10}{9} \div 2 = \frac{10}{9} \times \frac{1}{2} = \frac{10}{18} = \frac{5}{9} \]
Thus, the solution is:
\[ x = \frac{5}{9} \]
Therefore, the correct response is:
x equals Start Fraction 5 over 9 End Fraction.