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Simplify (x-5)/6 - 6/(x-5)

Mathematics

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Answer

(x−11)(x+1)6(x−5)\frac{(x-11)(x+1)}{6(x-5)}

Step-by-step solution

5 steps

  1. Find a common denominator for the two fractions.

    The denominators are 6 and x−5x-5. The least common multiple (LCM) of these is 6(x−5)6(x-5).

  2. Rewrite each fraction with the common denominator.

    x−56=(x−5)(x−5)6(x−5)=(x−5)26(x−5)\frac{x-5}{6} = \frac{(x-5)(x-5)}{6(x-5)} = \frac{(x-5)^2}{6(x-5)} 6x−5=6×66(x−5)=366(x−5)\frac{6}{x-5} = \frac{6 \times 6}{6(x-5)} = \frac{36}{6(x-5)}

  3. Subtract the rewritten fractions.

    (x−5)26(x−5)−366(x−5)=(x−5)2−366(x−5)\frac{(x-5)^2}{6(x-5)} - \frac{36}{6(x-5)} = \frac{(x-5)^2 - 36}{6(x-5)}

  4. Expand the numerator and simplify.

    Recognize that (x−5)2−36(x-5)^2 - 36 is a difference of squares, a2−b2=(a−b)(a+b)a^2 - b^2 = (a-b)(a+b), where a=(x−5)a = (x-5) and b=6b = 6.

    (x−5)2−36=((x−5)−6)((x−5)+6)(x-5)^2 - 36 = ((x-5) - 6)((x-5) + 6) =(x−5−6)(x−5+6)= (x-5-6)(x-5+6) =(x−11)(x+1)= (x-11)(x+1)

  5. Substitute the simplified numerator back into the expression.

    (x−11)(x+1)6(x−5)\frac{(x-11)(x+1)}{6(x-5)}
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