[administration]

Q. Value of the expression (x²-x+1)/(x-1) cannot lie between the interval ?

[Wonder Boy]

Whenever you encounter such a problem try to simplify it as far as possible and then accumulate the terms which can cancel out or give some known results

Lets try to simplify the expression  (x²-x+1)/(x-1).

So, (x²-x+1)/(x-1) = (x²-2x+1 +x )/(x-1) = [(x-1)² + x]/(x-1) = (x-1) + x/(x-1) = (x-1) + (x-1+1)/(x-1)

= (x-1)  + 1 + 1/(x-1)

Now we know that -2 <= y +1/y <= 2

–> -2 <= (x-1)  + 1/(x-1) <= 2

–> -1 <= (x-1) + 1 + 1/(x-1) <= 3

Hence the value of expression cant lie between (infinite, -1) U (3, infinite)

[Author]

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