1. **Problem:** Find the solution set of the inequality $$\frac{10 - 4x}{x^2 - 7x + 10} > 3$$. 2. **Step 1: Write the inequality clearly:** $$\frac{10 - 4x}{x^2 - 7x + 10} > 3$$ 3. **Step 2: Factor the denominator:** $$x^2 - 7x + 10 = (x - 5)(x - 2)$$ 4. **Step 3: Bring all terms to one side to form a single rational expression:** $$\frac{10 - 4x}{(x - 5)(x - 2)} - 3 > 0$$ 5. **Step 4: Express 3 as a fraction with the same denominator:** $$\frac{10 - 4x}{(x - 5)(x - 2)} - \frac{3(x - 5)(x - 2)}{(x - 5)(x - 2)} > 0$$ 6. **Step 5: Combine the fractions:** $$\frac{10 - 4x - 3(x - 5)(x - 2)}{(x - 5)(x - 2)} > 0$$ 7. **Step 6: Expand the numerator:** First expand $3(x - 5)(x - 2)$: $$3(x^2 - 7x + 10) = 3x^2 - 21x + 30$$ So numerator becomes: $$10 - 4x - (3x^2 - 21x + 30) = 10 - 4x - 3x^2 + 21x - 30 = -3x^2 + 17x - 20$$ 8. **Step 7: Rewrite the inequality:** $$\frac{-3x^2 + 17x - 20}{(x - 5)(x - 2)} > 0$$ 9. **Step 8: Factor the numerator if possible:** Multiply numerator by -1 to factor easier: $$3x^2 - 17x + 20$$ Try factors of $3 \times 20 = 60$ that sum to -17: -12 and -5. Rewrite: $$3x^2 - 12x - 5x + 20 = 3x(x - 4) - 5(x - 4) = (3x - 5)(x - 4)$$ So numerator: $$- (3x - 5)(x - 4)$$ 10. **Step 9: Rewrite the inequality with factored numerator:** $$\frac{- (3x - 5)(x - 4)}{(x - 5)(x - 2)} > 0$$ 11. **Step 10: Simplify the negative sign:** $$\frac{(3x - 5)(x - 4)}{(x - 5)(x - 2)} < 0$$ 12. **Step 11: Find critical points where numerator or denominator is zero:** - Numerator zeros: $3x - 5 = 0 \Rightarrow x = \frac{5}{3}$, and $x - 4 = 0 \Rightarrow x = 4$ - Denominator zeros: $x - 5 = 0 \Rightarrow x = 5$, and $x - 2 = 0 \Rightarrow x = 2$ 13. **Step 12: Determine intervals based on critical points:** $$(-\infty, \frac{5}{3}), (\frac{5}{3}, 2), (2, 4), (4, 5), (5, \infty)$$ 14. **Step 13: Test each interval for sign of the expression:** - Pick test points and check sign of numerator and denominator factors. 15. **Step 14: Exclude points where denominator is zero ($x=2,5$) since expression undefined there.** 16. **Step 15: Final solution set:** $$\boxed{\left( -\infty, \frac{5}{3} \right) \cup (2,4)}$$ --- **Answer:** The solution set is $$x \in \left( -\infty, \frac{5}{3} \right) \cup (2,4)$$.