Solve for x, 1÷(x-1)(x-2)+1÷(x-2)(x-3)+1÷(x-3)(x-4)=1÷6

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Solution

Take the LCM of the denominators = (x-1)(x-2)(x-3)(x-4)
Divide the LCM by the denominator of each term and multiply the result with the
numerator of the respective term.

(x-3)(x-4) + (x-1)(x-4) + (x-1)(x-2)
------------------------------------ = 1/6
(x-1)(x-2)(x-3)(x-4)

Use formula (x-a)(x-b) = x^2 - (a+b)x + ab

x^2 - 7x + 12 + x^2 - 5x + 4 + x^2 - 3x + 2
-------------------------------------------- = 1/6
(x-1)(x-2)(x-3)(x-4)

3x^2 - 15x + 18
--------------------- = 1/6
(x-1)(x-2)(x-3)(x-4)

3(x^2 -5x + 6)
------------------- = 1/6
(x-1)(x-2)(x-3)(x-4)

3(x - 2)(x - 3)
-------------------- = 1/6
(x-1)(x-2)(x-3)(x-4)

3
------------ = 1/6
(x-1)(x-4)

18 = (x-1)(x-4)
18 = x^2 - 5x + 4
x^2 - 5x - 14 = 0
(x - 7 )(x +2 ) = 0
x = 7 OR x = -2 .

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