Solve the equation: $\frac{x + 3}{x + 2} - \frac{x - 4}{x + 3} = \frac{x + 5}{x + 4} - \frac{x + 6}{x + 5}$

Created: 2 years ago | Updated: 1 year ago

Updated: 1 year ago

$\frac{x + 3}{x + 2} - \frac{x + 4}{x + 3} = \frac{x + 5}{x + 4} - \frac{x + 6}{x + 5} \Rightarrow \frac{x + 2 + 1}{x + 2} + \frac{1}{x + 2} - \frac{x + 3}{x + 3} - \frac{1}{x + 3} = \frac{x + 4 + 1}{x + 4} + \frac{x + 5 + 1}{x + 5} \Rightarrow \frac{x + 2}{x + 2} + \frac{1}{x + 2} - \frac{x + 3}{x + 3} - \frac{1}{x + 3} = \frac{x + 4}{x + 4} + \frac{1}{x + 4} - \frac{x + 5}{x + 5} - \frac{1}{x + 3} \Rightarrow 1 + \frac{1}{x + 2} - 1 - \frac{1}{x + 3} = 1 + \frac{1}{x + 4} - 1 - \frac{1}{x + 5} \Rightarrow \frac{1}{x + 2} - \frac{1}{x + 3} = \frac{1}{x + 4} - \frac{1}{x + 5} \Rightarrow \frac{x + 3 - (x + 2)}{(x + 2) (x + 3)} = \frac{x + 5 - (x + 4)}{(x + 4) (x + 5)} \Rightarrow \frac{x + 3 - x - 2}{(x + 2) (x + 3)} = \frac{x + 5 - x - 4}{(x + 4) (x + 5)} \Rightarrow \frac{1}{(x + 2) (x + 3)} = \frac{1}{(x + 4) (x + 5)} \Rightarrow (x + 4) (x + 5) = (x + 2) (x + 3) \Rightarrow x^{2} + 5 x + 4 x + 20 = x^{2} + 3 x + 2 x + 6 \Rightarrow x^{2} + 9 x - x^{2} - 5 x = 6 - 20 \Rightarrow 4 x = - 14 ∴ x = \frac{- 14}{4} = \frac{- 7}{2}$