1. **State the problem:** Solve the equation $37 \times 1 - \frac{1}{x} = \frac{6}{9}$ for $x$. 2. **Rewrite the equation:** Since $37 \times 1 = 37$, the equation becomes: $$37 - \frac{1}{x} = \frac{6}{9}$$ 3. **Simplify the fraction on the right:** $$\frac{6}{9} = \frac{2}{3}$$ 4. **Isolate the fraction term:** $$37 - \frac{1}{x} = \frac{2}{3} \implies -\frac{1}{x} = \frac{2}{3} - 37$$ 5. **Calculate the right side:** $$\frac{2}{3} - 37 = \frac{2}{3} - \frac{111}{3} = \frac{2 - 111}{3} = \frac{-109}{3}$$ 6. **Rewrite the equation:** $$-\frac{1}{x} = \frac{-109}{3}$$ 7. **Multiply both sides by $-1$ to simplify:** $$\cancel{-1} \times -\frac{1}{x} = \cancel{-1} \times \frac{-109}{3} \implies \frac{1}{x} = \frac{109}{3}$$ 8. **Solve for $x$ by taking the reciprocal:** $$x = \frac{3}{109}$$ **Final answer:** $$x = \frac{3}{109}$$