1. **State the problem:** Solve the equation $$|7(\alpha + \beta) + 4(\gamma + \delta)| = 7 \left| -12 + \frac{2}{7} + \left(-\frac{5}{2} - 1\right) \right|$$ for the expression inside the absolute values. 2. **Simplify the right side inside the absolute value:** Calculate the sum inside the absolute value on the right: $$-12 + \frac{2}{7} + \left(-\frac{5}{2} - 1\right) = -12 + \frac{2}{7} - \frac{5}{2} - 1$$ 3. **Find a common denominator and combine terms:** Convert all terms to fractions with denominator 14: $$-12 = -\frac{168}{14}, \quad \frac{2}{7} = \frac{4}{14}, \quad -\frac{5}{2} = -\frac{35}{14}, \quad -1 = -\frac{14}{14}$$ Sum: $$-\frac{168}{14} + \frac{4}{14} - \frac{35}{14} - \frac{14}{14} = \frac{-168 + 4 - 35 - 14}{14} = \frac{-213}{14}$$ 4. **Calculate the absolute value:** $$\left| -12 + \frac{2}{7} + \left(-\frac{5}{2} - 1\right) \right| = \left| \frac{-213}{14} \right| = \frac{213}{14}$$ 5. **Multiply by 7:** $$7 \times \frac{213}{14} = \frac{7 \times 213}{14} = \frac{1491}{14}$$ 6. **Rewrite the original equation:** $$|7(\alpha + \beta) + 4(\gamma + \delta)| = \frac{1491}{14}$$ 7. **Interpretation:** The absolute value expression on the left equals $$\frac{1491}{14}$$. Without additional information about $$\alpha, \beta, \gamma, \delta$$, this is the simplified form of the equation. **Final answer:** $$|7(\alpha + \beta) + 4(\gamma + \delta)| = \frac{1491}{14}$$