1. **State the problem:** Solve the inequality $$\frac{10 - 4x}{x^2 - 7x + 10} > 3$$. 2. **Rewrite the inequality:** Move all terms to one side to compare to zero: $$\frac{10 - 4x}{x^2 - 7x + 10} - 3 > 0$$ 3. **Find a common denominator and combine:** $$\frac{10 - 4x - 3(x^2 - 7x + 10)}{x^2 - 7x + 10} > 0$$ 4. **Expand the numerator:** $$10 - 4x - 3x^2 + 21x - 30 = -3x^2 + 17x - 20$$ 5. **Rewrite the inequality:** $$\frac{-3x^2 + 17x - 20}{x^2 - 7x + 10} > 0$$ 6. **Factor numerator and denominator:** - Denominator: $$x^2 - 7x + 10 = (x - 5)(x - 2)$$ - Numerator: Find factors of $$-3x^2 + 17x - 20$$ Try factoring numerator: $$-3x^2 + 17x - 20 = -(3x^2 - 17x + 20)$$ Find factors of $$3x^2 - 17x + 20$$: Try $$(3x - 5)(x - 4) = 3x^2 - 12x - 5x + 20 = 3x^2 - 17x + 20$$ So numerator: $$-(3x - 5)(x - 4)$$ 7. **Rewrite inequality with factors:** $$\frac{-(3x - 5)(x - 4)}{(x - 5)(x - 2)} > 0$$ 8. **Simplify by factoring out the negative sign:** $$\frac{-(3x - 5)(x - 4)}{(x - 5)(x - 2)} = - \frac{(3x - 5)(x - 4)}{(x - 5)(x - 2)}$$ So inequality is: $$- \frac{(3x - 5)(x - 4)}{(x - 5)(x - 2)} > 0$$ 9. **Multiply both sides by -1 (flip inequality):** $$\frac{(3x - 5)(x - 4)}{(x - 5)(x - 2)} < 0$$ 10. **Identify critical points:** - Numerator zeros: $$3x - 5 = 0 \Rightarrow x = \frac{5}{3}$$, $$x - 4 = 0 \Rightarrow x = 4$$ - Denominator zeros (excluded from domain): $$x = 2, 5$$ 11. **Determine sign intervals:** Intervals to test: $$(-\infty, 2), (2, \frac{5}{3}), (\frac{5}{3}, 4), (4, 5), (5, \infty)$$ Note $$\frac{5}{3} \approx 1.666$$ so order is $$(-\infty, \frac{5}{3}), (\frac{5}{3}, 2), (2, 4), (4, 5), (5, \infty)$$ 12. **Test each interval:** - For $$x < \frac{5}{3}$$ (e.g., 1): numerator factors: $(3(1)-5) = -2$ (negative), $(1-4) = -3$ (negative), numerator positive (negative*negative). Denominator: $(1-5) = -4$ (negative), $(1-2) = -1$ (negative), denominator positive. Fraction positive, but inequality requires fraction < 0, so no. - For $$\frac{5}{3} < x < 2$$ (e.g., 1.8): numerator: $(3(1.8)-5) = 0.4$ (positive), $(1.8-4) = -2.2$ (negative), numerator negative. Denominator: $(1.8-5) = -3.2$ (negative), $(1.8-2) = -0.2$ (negative), denominator positive. Fraction negative < 0, satisfies inequality. - For $$2 < x < 4$$ (e.g., 3): numerator: $(3(3)-5) = 4$ (positive), $(3-4) = -1$ (negative), numerator negative. Denominator: $(3-5) = -2$ (negative), $(3-2) = 1$ (positive), denominator negative. Fraction negative/negative = positive > 0, no. - For $$4 < x < 5$$ (e.g., 4.5): numerator: $(3(4.5)-5) = 8.5$ (positive), $(4.5-4) = 0.5$ (positive), numerator positive. Denominator: $(4.5-5) = -0.5$ (negative), $(4.5-2) = 2.5$ (positive), denominator negative. Fraction positive/negative = negative < 0, satisfies inequality. - For $$x > 5$$ (e.g., 6): numerator: $(3(6)-5) = 13$ (positive), $(6-4) = 2$ (positive), numerator positive. Denominator: $(6-5) = 1$ (positive), $(6-2) = 4$ (positive), denominator positive. Fraction positive > 0, no. 13. **Domain restrictions:** Exclude $$x=2$$ and $$x=5$$ because denominator zero. 14. **Final solution set:** $$\left(\frac{5}{3}, 2\right) \cup (4, 5)$$ **Answer:** The solution set is $$\boxed{\left(\frac{5}{3}, 2\right) \cup (4, 5)}$$.