How do you multiply (2x) / (x+2) - 2 = (x-8) / (x-2)?

Aug 12, 2018

$x = - 4 \mathmr{and} x = 6$

Explanation:

$\frac{2 x}{x + 2} - 2 = \frac{x - 8}{x - 2}$

$\therefore = \frac{2 x \left(x - 2\right) - 2 \left(x + 2\right) \left(x - 2\right) = \left(x + 2\right) \left(x - 8\right)}{\left(x + 2\right) \left(x - 2\right)}$

Multiply both sides by $\left(x + 2\right) \left(x - 2\right)$

$\therefore = 2 x \left(x - 2\right) - 2 \left(x + 2\right) \left(x - 2\right) = \left(x + 2\right) \left(x - 8\right)$

$\therefore = 2 {x}^{2} - 4 x - 2 {x}^{2} + 8 = {x}^{2} - 6 x - 16$

$\therefore = 2 {x}^{2} - {x}^{2} - 2 {x}^{2} - 4 x + 6 x + 8 + 16 = 0$

$\therefore = - {x}^{2} + 2 x + 24 = 0$

multiply both sides by$- 1$

$\therefore = {x}^{2} - 2 x - 24 = 0$

$\therefore = \left(x + 4\right) \left(x - 6\right) = 0$

$\therefore = x = - 4 \mathmr{and} x = 6$

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check:-

substitute $x = - 4$

$\therefore = \frac{2 \left(- 4\right)}{\left(- 4\right) + 2} - 2 = \frac{\left(- 4\right) - 8}{\left(- 4\right) - 2}$

$\therefore = \frac{- 8}{-} 2 - 2 = \frac{- 12}{-} 6$

$\therefore = 4 - 2 = 2$

substitute $x = 6$

$\frac{2 \left(6\right)}{\left(6\right) + 2} - 2 = \frac{\left(6\right) - 8}{\left(6 - 2\right)}$

$\therefore = \frac{12}{8} - 2 = \frac{- 2}{4}$

$\therefore = \frac{3}{2} - 2 = - \frac{1}{2}$

$1 \frac{1}{2} - 2 = - \frac{1}{2}$

$\therefore = - \frac{1}{2} = - \frac{1}{2}$