1. **State the problem:** Solve the expression $$\left(\frac{3}{2} - \frac{23}{42} + \frac{8}{21}\right) = \left(\frac{17}{6} - \frac{29}{15}\right) \cdot \left(\frac{5}{2} - \frac{17}{15} - \frac{13}{30}\right)$$ for the value of the left side and the right side. 2. **Simplify the left side:** Find a common denominator for $$\frac{3}{2}, \frac{23}{42}, \frac{8}{21}$$. The least common denominator (LCD) is 42. Convert each fraction: $$\frac{3}{2} = \frac{3 \times 21}{2 \times 21} = \frac{63}{42}$$ $$\frac{23}{42}$$ stays the same. $$\frac{8}{21} = \frac{8 \times 2}{21 \times 2} = \frac{16}{42}$$ Now sum: $$\frac{63}{42} - \frac{23}{42} + \frac{16}{42} = \frac{63 - 23 + 16}{42} = \frac{56}{42}$$ Simplify: $$\frac{56}{42} = \frac{\cancel{56}^{8} \times 7}{\cancel{42}^{6} \times 7} = \frac{8}{6} = \frac{4}{3}$$ 3. **Simplify the first term on the right side:** $$\frac{17}{6} - \frac{29}{15}$$ Find LCD of 6 and 15, which is 30. Convert: $$\frac{17}{6} = \frac{17 \times 5}{6 \times 5} = \frac{85}{30}$$ $$\frac{29}{15} = \frac{29 \times 2}{15 \times 2} = \frac{58}{30}$$ Subtract: $$\frac{85}{30} - \frac{58}{30} = \frac{27}{30}$$ Simplify: $$\frac{27}{30} = \frac{\cancel{27}^{9} \times 3}{\cancel{30}^{10} \times 3} = \frac{9}{10}$$ 4. **Simplify the second term on the right side:** $$\frac{5}{2} - \frac{17}{15} - \frac{13}{30}$$ Find LCD of 2, 15, and 30, which is 30. Convert: $$\frac{5}{2} = \frac{5 \times 15}{2 \times 15} = \frac{75}{30}$$ $$\frac{17}{15} = \frac{17 \times 2}{15 \times 2} = \frac{34}{30}$$ $$\frac{13}{30}$$ stays the same. Calculate: $$\frac{75}{30} - \frac{34}{30} - \frac{13}{30} = \frac{75 - 34 - 13}{30} = \frac{28}{30}$$ Simplify: $$\frac{28}{30} = \frac{\cancel{28}^{14} \times 2}{\cancel{30}^{15} \times 2} = \frac{14}{15}$$ 5. **Multiply the two simplified terms on the right side:** $$\frac{9}{10} \times \frac{14}{15} = \frac{9 \times 14}{10 \times 15} = \frac{126}{150}$$ Simplify: $$\frac{126}{150} = \frac{\cancel{126}^{42} \times 3}{\cancel{150}^{50} \times 3} = \frac{42}{50} = \frac{\cancel{42}^{21} \times 2}{\cancel{50}^{25} \times 2} = \frac{21}{25}$$ 6. **Compare both sides:** Left side = $$\frac{4}{3} = \frac{100}{75}$$ (converted to denominator 75) Right side = $$\frac{21}{25} = \frac{63}{75}$$ (converted to denominator 75) Since $$\frac{100}{75} \neq \frac{63}{75}$$, the two sides are not equal. **Final answer:** $$\left(\frac{3}{2} - \frac{23}{42} + \frac{8}{21}\right) = \frac{4}{3}$$ $$\left(\frac{17}{6} - \frac{29}{15}\right) \cdot \left(\frac{5}{2} - \frac{17}{15} - \frac{13}{30}\right) = \frac{21}{25}$$ They are not equal.