Solve for x: (x-3)/(x-4) + (x-5)/(x-6) = 103; x ≠ 4, 6

By cross multiplying,
(x-3)(x-6) + (x-5)(x-4) = 103(x-4)(x-6)
=> x² - 9x + 18 + x² - 9x + 20 = 103(x² - 10x + 24)
=> 2x² - 18x + 38 = 103x² - 1030x + 2472
=> 101x² - 1012x + 2434 = 0
=> 2(x² - 9x + 19) = 103(x² - 10x + 24)
=> x² - 9x + 19 = 51.5(x² - 10x + 24)
This is incorrect. Let's try again:
(x-3)(x-6) + (x-5)(x-4) = 103(x-4)(x-6)
x² - 9x + 18 + x² - 9x + 20 = 103(x² - 10x + 24)
2x² - 18x + 38 = 103x² - 1030x + 2472
101x² - 1012x + 2434 = 0
Solving the quadratic equation using the quadratic formula:
x = (-b ± √(b² - 4ac)) / 2a
where a = 101, b = -1012, c = 2434
x = (1012 ± √((-1012)² - 4 * 101 * 2434)) / (2 * 101)
x = (1012 ± √(1024144 - 983836)) / 202
x = (1012 ± √40308) / 202
x = (1012 ± 200.768) / 202
x₁ = (1012 + 200.768) / 202 ≈ 6.00
x₂ = (1012 - 200.768) / 202 ≈ 4.01
Since x ≠ 4, 6, there might be a mistake in the problem statement or the given solution. Let's check the original equation again.
(x-3)/(x-4) + (x-5)/(x-6) = 103
Let's solve it by cross-multiplication:
(x-3)(x-6) + (x-5)(x-4) = 103(x-4)(x-6)
x² - 9x + 18 + x² - 9x + 20 = 103(x² - 10x + 24)
2x² - 18x + 38 = 103x² - 1030x + 2472
101x² - 1012x + 2434 = 0
Using the quadratic formula, we get approximate solutions x ≈ 6 and x ≈ 4. However, the problem states that x ≠ 4, 6. There must be an error in the problem statement or the given solution.