Simplify the fraction (3+ 1/x)/(9-1/x^2)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

We have to simplify: (3+ 1/x)/(9-1/x^2)

(3+ 1/x)/(9-1/x^2)

=> (3 + 1/x)/(3 - 1/x)(3 + 1/x)

=> 1/(3 - 1/x)

=> x/(3x - 1)

The required simplified form is x/(3x - 1)

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

We'll multiply both numerator and denominator by the largest denominator, namely x^2:
(3x^2 + x^2/x)/(9x^2 - x^2/x^2) = (3x^2 + x)/(9x^2 - 1)
The difference of two squares from denominator returns the product:
(9x^2 - 1) = (3x-1)(3x+1)
We'll factorize the numerator by x:
(3x^2 + x)/(9x^2 - 1) = x(3x+1)/(3x-1)(3x+1)
We'll divide the fraction by (3x+1):
x(3x+1)/(3x-1)(3x+1) = x/(3x-1)
Therefore, simplifying the given fraction, we'll get: x/(3x-1).

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