1. **State the problem:** Solve the inequality $$\frac{x^2 - 3x}{2x - 4} \leq 5$$. 2. **Rewrite the inequality:** Bring all terms to one side to get a single rational expression. $$\frac{x^2 - 3x}{2x - 4} - 5 \leq 0$$ 3. **Find a common denominator and combine:** $$\frac{x^2 - 3x - 5(2x - 4)}{2x - 4} \leq 0$$ Simplify numerator: $$x^2 - 3x - 10x + 20 = x^2 - 13x + 20$$ So inequality becomes: $$\frac{x^2 - 13x + 20}{2x - 4} \leq 0$$ 4. **Factor numerator and denominator:** Numerator: $$x^2 - 13x + 20 = (x - 5)(x - 4)$$ Denominator: $$2x - 4 = 2(x - 2)$$ 5. **Rewrite inequality:** $$\frac{(x - 5)(x - 4)}{2(x - 2)} \leq 0$$ 6. **Identify critical points:** Points where numerator or denominator is zero: $$x = 5, 4, 2$$ 7. **Determine intervals for testing:** $$(-\infty, 2), (2, 4), (4, 5), (5, \infty)$$ 8. **Check sign of expression on each interval:** - For $$x < 2$$ (e.g., $$x=0$$): Numerator is positive (since (0-5)(0-4) = 20), denominator is negative (2(0-2) = -4), so fraction is negative. - For $$2 < x < 4$$ (e.g., $$x=3$$): Numerator (3-5)(3-4) = (-2)(-1) = 2 positive; denominator 2(3-2) = 2 positive; fraction positive. - For $$4 < x < 5$$ (e.g., $$x=4.5$$): Numerator (4.5-5)(4.5-4) = (-0.5)(0.5) = -0.25 negative; denominator 2(4.5-2) = 5 positive; fraction negative. - For $$x > 5$$ (e.g., $$x=6$$): Numerator (6-5)(6-4) = (1)(2) = 2 positive; denominator 2(6-2) = 8 positive; fraction positive. 9. **Include points where expression equals zero:** Numerator zero at $$x=4, 5$$ (fraction equals zero). Denominator zero at $$x=2$$ (undefined - vertical asymptote). 10. **Construct solution from intervals where expression is $$\leq 0$$:** These are intervals where fraction is negative or zero: - $$(-\infty, 2)$$ (negative) - $$[4, 5]$$ (zero at endpoints, negative between) 11. **Check domain restriction:** Exclude $$x=2$$ from solution set since denominator is zero. 12. **Final solution:** $$(-\infty, 2) \cup [4, 5]$$ with $$x \neq 2$$. **Answer:** $$\boxed{(-\infty, 2) \cup [4, 5]}$$ with exclusion of $$x=2$$ which is not in the domain.