1. **State the problem:** Solve the equation $$\frac{6}{x} + 6 = \frac{4}{7}$$ for $x$. 2. **Isolate the fraction term:** Subtract 6 from both sides: $$\frac{6}{x} = \frac{4}{7} - 6$$ 3. **Simplify the right side:** Convert 6 to a fraction with denominator 7: $$6 = \frac{42}{7}$$ So, $$\frac{6}{x} = \frac{4}{7} - \frac{42}{7} = \frac{4 - 42}{7} = \frac{-38}{7}$$ 4. **Rewrite the equation:** $$\frac{6}{x} = \frac{-38}{7}$$ 5. **Cross multiply to solve for $x$:** $$6 \times 7 = -38 \times x$$ $$42 = -38x$$ 6. **Divide both sides by -38:** $$x = \frac{42}{-38}$$ Show cancellation: $$x = \frac{\cancel{42}}{\cancel{38}} \times \frac{2}{2} = \frac{21}{-19} = -\frac{21}{19}$$ 7. **Final answer:** $$x = -\frac{21}{19}$$