1. **State the problem:** Solve the equation $$\frac{45 - 2x}{15} - \frac{4x + 10}{5} = \frac{15 - 14x}{9}$$ for $x$. 2. **Identify the formula and rules:** To solve equations with fractions, find a common denominator or multiply both sides by the least common multiple (LCM) of denominators to clear fractions. 3. **Find the LCM of denominators:** The denominators are 15, 5, and 9. - Prime factors: 15 = 3 \times 5, 5 = 5, 9 = 3^2 - LCM = 3^2 \times 5 = 9 \times 5 = 45 4. **Multiply both sides of the equation by 45 to clear denominators:** $$45 \times \left(\frac{45 - 2x}{15} - \frac{4x + 10}{5}\right) = 45 \times \frac{15 - 14x}{9}$$ 5. **Distribute multiplication:** $$45 \times \frac{45 - 2x}{15} = 3(45 - 2x) = 135 - 6x$$ $$45 \times \frac{4x + 10}{5} = 9(4x + 10) = 36x + 90$$ $$45 \times \frac{15 - 14x}{9} = 5(15 - 14x) = 75 - 70x$$ 6. **Rewrite the equation:** $$135 - 6x - (36x + 90) = 75 - 70x$$ 7. **Simplify the left side:** $$135 - 6x - 36x - 90 = 75 - 70x$$ $$135 - 90 - 6x - 36x = 75 - 70x$$ $$45 - 42x = 75 - 70x$$ 8. **Bring variables to one side and constants to the other:** Add $70x$ to both sides: $$45 - 42x + 70x = 75 - 70x + 70x$$ $$45 + 28x = 75$$ Subtract 45 from both sides: $$45 + 28x - 45 = 75 - 45$$ $$28x = 30$$ 9. **Solve for $x$:** $$x = \frac{30}{28}$$ Simplify the fraction by dividing numerator and denominator by 2: $$x = \frac{\cancel{30}{}^{15}}{\cancel{28}{}^{14}}$$ 10. **Final answer:** $$x = \frac{15}{14}$$