Question

Solve for x: 1/(x-1)(x-2) + 1/(x-2)(x-3) + 1/(x-3)(x-4) = 1/16.

Answer:

x = -2,7

Taking LCM of the first two terms:
(x-3)/(x-1)(x-2)(x-3) + (x-1)/(x-1)(x-2)(x-3) + 1/(x-3)(x-4) = 1/16
⇒(x-3+x-1)/(x-1)(x-2)(x-3) + 1/(x-3)(x-4) = 1/16
⇒(2x-4)/(x-1)(x-2)(x-3) + 1/(x-3)(x-4) = 1/16
⇒2(x-2)/(x-1)(x-2)(x-3) + 1/(x-3)(x-4) = 1/16
⇒2/(x-1)(x-3) + 1/(x-3)(x-4) = 1/16
Now, taking LCM of the remaining terms:
2(x-4)/(x-1)(x-3)(x-4) + (x-1)/(x-1)(x-3)(x-4) = 1/16
⇒(2x-8+x-1)/(x-1)(x-3)(x-4) = 1/16
⇒(3x-9)/(x-1)(x-3)(x-4) = 1/16
⇒3(x-3)/(x-1)(x-3)(x-4) = 1/16
⇒3/(x-1)(x-4) = 1/16
⇒(x-1)(x-4) = 18
⇒x²-5x+4 = 18
⇒x²-5x-14 = 0
⇒x²-7x+2x-14 = 0
⇒x(x-7)+2(x-7) = 0
⇒(x+2)(x-7) = 0
Therefore, either x+2 = 0 in which case x will be equal to -2 or x-7 = 0 in which x will be equal to 7.
Therefore, the vaule of x can be either -2 or 7.



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