# Chapter 7 - Section 7.7 - Simplifying Complex Fractions - Exercise Set - Page 549: 39

$\dfrac{\dfrac{6}{x-5}+\dfrac{x}{x-2}}{\dfrac{3}{x-6}-\dfrac{2}{x-5}}=\dfrac{(x+4)(x-6)}{x-2}$

#### Work Step by Step

$\dfrac{\dfrac{6}{x-5}+\dfrac{x}{x-2}}{\dfrac{3}{x-6}-\dfrac{2}{x-5}}$ Evaluate the sum indicated in the numerator and the substraction indicated in the denominator: $\dfrac{\dfrac{6}{x-5}+\dfrac{x}{x-2}}{\dfrac{3}{x-6}-\dfrac{2}{x-5}}=\dfrac{\dfrac{6(x-2)+x(x-5)}{(x-5)(x-2)}}{\dfrac{3(x-5)-2(x-6)}{(x-6)(x-5)}}=...$ $...=\dfrac{\dfrac{6x-12+x^{2}-5x}{(x-5)(x-2)}}{\dfrac{3x-15-2x+12}{(x-6)(x-5)}}=\dfrac{\dfrac{x^{2}+x-12}{(x-5)(x-2)}}{\dfrac{x-3}{(x-6)(x-5)}}=...$ Evaluate the division: $...=\dfrac{x^{2}+x-12}{(x-5)(x-2)}\div\dfrac{x-3}{(x-6)(x-5)}=...$ $...=\dfrac{(x^{2}+x-12)(x-6)(x-5)}{(x-5)(x-2)(x-3)}=...$ Factor the first parentheses in the numerator and simplify: $...=\dfrac{(x+4)(x-3)(x-6)(x-5)}{(x-5)(x-2)(x-3)}=...$ $...=\dfrac{(x+4)(x-6)}{x-2}$

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