Solve for x : (((3x − 28)/(3x − 26)))^3 = ((x − 10)/(x − 8))


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Question Number 181144 by Agnibhoo98 last updated on 22/Nov/22

$Solve for x :$ ${(\frac{3 x - 28}{3 x - 26})}^{3} = \frac{x - 10}{x - 8}$
	Answered by Rasheed.Sindhi last updated on 22/Nov/22

	${(\frac{3 x - 28}{3 x - 26})}^{3} = \frac{x - 10}{x - 8}$ $\frac{{(3 x - 27 - 1)}^{3}}{{(3 x - 27 + 1)}^{3}} = \frac{x - 9 - 1}{x - 9 + 1}$ $\frac{{(3 (x - 9) - 1)}^{3}}{{(3 (x - 9) + 1)}^{3}} = \frac{x - 9 - 1}{x - 9 + 1}$ $x - 9 \to y :$ $\frac{{(3 y - 1)}^{3}}{{(3 y + 1)}^{3}} = \frac{y - 1}{y + 1}$ $\frac{{(3 y - 1)}^{2}}{{(3 y + 1)}^{2}} = \frac{(3 y + 1) (y - 1)}{(3 y - 1) (y + 1)}$ $\frac{9 y^{2} - 6 y + 1}{9 y^{2} + 6 y + 1} - 1 = \frac{3 y^{2} - 2 y - 1}{3 y^{2} + 2 y - 1} - 1$ $\frac{- 12 y}{9 y^{2} + 6 y + 1} = \frac{- 4 y}{3 y^{2} + 2 y - 1}$ $\frac{3 y}{9 y^{2} + 6 y + 1} = \frac{y}{3 y^{2} + 2 y - 1}$ $9 y^{3} + 6 y^{2} - 3 y = 9 y^{3} + 6 y^{2} + y$ $- 4 y = 0$ $y = 0$ $x - 9 = 0$ $x = 9$
	Answered by Rasheed.Sindhi last updated on 22/Nov/22

	${(\frac{3 x - 28}{3 x - 26})}^{3} = \frac{x - 10}{x - 8}$ ${(\frac{3 x - 28}{3 x - 26})}^{2} (\frac{3 x - 28}{3 x - 26}) = \frac{x - 10}{x - 8}$ $\frac{{(3 x - 28)}^{2}}{{(3 x - 26)}^{2}} = \frac{3 x - 26}{3 x - 28} \cdot \frac{x - 10}{x - 8}$ $\frac{9 x^{2} - 168 x + 784}{9 x^{2} - 156 x + 676} = \frac{3 x^{2} - 56 x + 260}{3 x^{2} - 52 x + 224}$ $\frac{9 x^{2} - 168 x + 784}{9 x^{2} - 156 x + 676} - 1 = \frac{3 x^{2} - 56 x + 260}{3 x^{2} - 52 x + 224} - 1$ $\frac{- 12 x + 108}{9 x^{2} - 156 x + 676} = \frac{- 4 x + 36}{3 x^{2} - 52 x + 224}$ $\frac{3 x - 27}{9 x^{2} - 156 x + 676} = \frac{x - 9}{3 x^{2} - 52 x + 224}$ $\frac{3 (x - 9)}{9 x^{2} - 156 x + 676} - \frac{x - 9}{3 x^{2} - 52 x + 224} = 0$ $(x - 9) (\frac{3}{9 x^{2} - 156 x + 676} - \frac{1}{3 x^{2} - 52 x + 224}) = 0$ $\Rightarrow x - 9 = 0 \Rightarrow x = 9$ $\Rightarrow \frac{3}{9 x^{2} - 156 x + 676} = \frac{1}{3 x^{2} - 52 x + 224}$ $9 x^{2} - 156 x + 672 = 9 x^{2} - 156 x + 676$ $Absurd$
	Answered by Rasheed.Sindhi last updated on 23/Nov/22

	$\frac{{(3 x - 28)}^{3}}{{(3 x - 26)}^{3}} = \frac{x - 10}{x - 8}$ $\frac{{(3 x - 28)}^{3}}{{(3 x - 26)}^{3}} = \frac{3 x - 30}{3 x - 24}$ $3 x - 28 = y$ $3 x - 26 = 3 x - 28 + 2 = y + 2$ $3 x - 30 = 3 x - 28 - 2 = y - 2$ $3 x - 24 = 3 x - 28 + 4 = y + 4$ $\frac{y^{3}}{{(y + 2)}^{3}} = \frac{y - 2}{y + 4}$ $\frac{y^{2}}{{(y + 2)}^{2}} = \frac{(y + 2) (y - 2)}{y (y + 4)}$ $\frac{y^{2}}{y^{2} + 4 y + 4} - 1 = \frac{y^{2} - 4}{y^{2} + 4 y} - 1$ $\frac{- 4 y - 4}{y^{2} + 4 y + 4} = \frac{- 4 y - 4}{y^{2} + 4 y}$ $\frac{y + 1}{y^{2} + 4 y + 4} - \frac{y + 1}{y^{2} + 4 y} = 0$ $y^{3} + 4 y^{2} + 4 y + y^{2} + 4 y + 4 = y^{3} + 4 y^{2} + y^{2} + 4 y$ $y^{3} + 5 y^{2} + 8 y + 4 = y^{3} + 5 y^{2} + 4 y$ $4 y = - 4$ $y = - 1$ $\Rightarrow 3 x - 28 = - 1 \Rightarrow x = 9$
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