Question #cf38f

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Question #cf38f

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Answer 1 · 2017-03-17T10:05:16

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{x - 3}{x - 4} - \frac{x - 7}{x - 8}$ $(x-1)/(x-2) -(x-5)/(x-6)=(x-3)/(x-4)-(x-7)/(x-8)$

$\Rightarrow \frac{x - 2 + 1}{x - 2} - \frac{x - 6 + 1}{x - 6} = \frac{x - 4 + 1}{x - 4} - \frac{x - 8 + 1}{x - 8}$ $=>(x-2+1)/(x-2) -(x-6+1)/(x-6)=(x-4+1)/(x-4)-(x-8+1)/(x-8)$

$\Rightarrow 1 + \frac{1}{x - 2} - 1 - \frac{1}{x - 6} = 1 + \frac{1}{x - 4} - 1 - \frac{1}{x - 8}$ $=>1+1/(x-2) -1-1/(x-6)=1+1/(x-4)-1-1/(x-8)$

$\Rightarrow \frac{1}{x - 2} - \frac{1}{x - 6} = \frac{1}{x - 4} - \frac{1}{x - 8}$ $=>1/(x-2) -1/(x-6)=1/(x-4)-1/(x-8)$

$\Rightarrow \frac{x - 6 - x + 2}{(x - 2) (x - 6)} = \frac{x - 8 - x + 4}{(x - 4) (x - 8)}$ $=>(x-6-x+2)/((x-2)(x-6))=(x-8-x+4)/((x-4)(x-8))$

$\Rightarrow - \frac{4}{(x - 2) (x - 6)} = - \frac{4}{(x - 4) (x - 8)}$ $=>-4/((x-2)(x-6))=-4/((x-4)(x-8))$

$\Rightarrow (x - 2) (x - 6) = (x - 4) (x - 8)$ $=>(x-2)(x-6)=(x-4)(x-8)$

$\Rightarrow x^{2} - 8 x + 12 = x^{2} - 12 x + 32$ $=>x^2-8x+12=x^2-12x+32$

$\Rightarrow 12 x - 8 x = 32 - 12 = 20$ $=>12x-8x=32-12=20$

$\Rightarrow 4 x = 20$ $=>4x=20$

$\Rightarrow x = \frac{20}{4} = 5$ $=>x=20/4=5$

Answer 2 · 2017-03-17T10:12:11

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{(x - 1) (x - 6) - (x - 2) (x - 5)}{(x - 2) (x - 6)} = \frac{x^{2} - 6 x - x + 6 - x^{2} + 5 x + 2 x - 10}{(x - 2) (x - 6)} = - \frac{4}{(x - 2) (x - 6)}$ $(x-1)/(x-2) - (x-5)/(x-6) = [(x-1)(x-6)-(x-2)(x-5)] /[(x-2)(x-6)] = (x^2 -6x -x + 6 - x^2 + 5x + 2x - 10)/[(x-2)(x-6)] =-4/[(x-2)(x-6)]$

Explanation:

$\frac{x - 1}{x - 2} - \frac{x - 5}{x - 6} = \frac{(x - 1) (x - 6) - (x - 2) (x - 5)}{(x - 2) (x - 6)} = \frac{x^{2} - 6 x - x + 6 - x^{2} + 5 x + 2 x - 10}{(x - 2) (x - 6)} = - \frac{4}{(x - 2) (x - 6)}$ $(x-1)/(x-2) - (x-5)/(x-6) = [(x-1)(x-6)-(x-2)(x-5)] /[(x-2)(x-6)]= (x^2 -6x -x + 6 - x^2 + 5x + 2x - 10)/[(x-2)(x-6)] =-4/[(x-2)(x-6)]$
$\frac{x - 3}{x - 4} - \frac{x - 7}{x - 8} = \frac{(x - 3) (x - 8) - (x - 4) (x - 7)}{(x - 4) (x - 8)} = \frac{x^{2} - 8 x - 3 x + 24 - x^{2} + 7 x + 4 x - 28}{(x - 4) (x - 8)} = - \frac{4}{(x - 4) (x - 8)}$ $(x-3)/(x-4) - (x-7)/(x-8) = [(x-3)(x-8)-(x-4)(x-7)] /[(x-4)(x-8)]= (x^2 -8x -3x + 24 - x^2 + 7x + 4x - 28)/[(x-4)(x-8)] =-4/[(x-4)(x-8)]$
So you get
$(x - 2) (x - 6) = (x - 4) (x - 8)$ $(x-2)(x-6) = (x-4)(x-8)$
$4 x = 20 .$ $4x=20.$
$x = 5 .$ $x = 5.$

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