1. **State the problem:** Solve the inequality $$\frac{x-1}{5} \geq x - \frac{2}{3}$$. 2. **Formula and rules:** To solve inequalities involving fractions, first eliminate denominators by multiplying both sides by the least common denominator (LCD). Remember, when multiplying or dividing by a negative number, reverse the inequality sign. 3. **Find the LCD:** The denominators are 5 and 3, so the LCD is 15. 4. **Multiply both sides by 15:** $$15 \times \frac{x-1}{5} \geq 15 \times \left(x - \frac{2}{3}\right)$$ 5. **Simplify each term:** $$3(x-1) \geq 15x - 10$$ 6. **Distribute:** $$3x - 3 \geq 15x - 10$$ 7. **Bring all terms to one side:** $$3x - 3 - 15x + 10 \geq 0$$ 8. **Combine like terms:** $$-12x + 7 \geq 0$$ 9. **Isolate x:** $$-12x \geq -7$$ 10. **Divide both sides by -12, reversing the inequality:** $$x \leq \frac{\cancel{-7}}{\cancel{-12}} = \frac{7}{12}$$ **Final answer:** $$x \leq \frac{7}{12}$$