1. **State the problem:** Solve the equation $$- \frac{5}{4} x = \frac{10}{20}$$ for $x$. 2. **Simplify the right side:** $$\frac{10}{20} = \frac{1}{2}$$. 3. **Rewrite the equation:** $$- \frac{5}{4} x = \frac{1}{2}$$. 4. **Isolate $x$ by dividing both sides by $- \frac{5}{4}$:** $$x = \frac{\frac{1}{2}}{- \frac{5}{4}}$$ 5. **Dividing by a fraction is the same as multiplying by its reciprocal:** $$x = \frac{1}{2} \times \left(- \frac{4}{5}\right)$$ 6. **Multiply the numerators and denominators:** $$x = - \frac{4}{10}$$ 7. **Simplify the fraction by dividing numerator and denominator by 2:** $$x = - \frac{\cancel{4}^{2}}{\cancel{10}^{5}} = - \frac{2}{5}$$ **Final answer:** $$x = - \frac{2}{5}$$