1. **State the problem:** Solve the system of inequalities: $$3x \geq 2$$ and $$-2x \geq -20$$ 2. **Solve the first inequality:** $$3x \geq 2$$ Divide both sides by 3 (positive number, inequality direction stays the same): $$x \geq \frac{2}{3}$$ 3. **Solve the second inequality:** $$-2x \geq -20$$ Divide both sides by -2 (negative number, inequality direction reverses): $$x \leq 10$$ 4. **Combine the solutions:** The solution must satisfy both inequalities simultaneously: $$\frac{2}{3} \leq x \leq 10$$ 5. **Interpretation:** The values of $x$ that satisfy both inequalities are all real numbers between $\frac{2}{3}$ and $10$, inclusive. **Final answer:** $$\boxed{\frac{2}{3} \leq x \leq 10}$$