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Question
Solve for x: \[\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}, x \neq 3, - 5\]
Solution
\[\frac{1}{x - 3} - \frac{1}{x + 5} = \frac{1}{6}\]
\[ \Rightarrow \frac{x + 5 - x + 3}{\left( x - 3 \right)\left( x + 5 \right)} = \frac{1}{6}\]
\[ \Rightarrow \frac{8}{\left( x - 3 \right)\left( x + 5 \right)} = \frac{1}{6}\]
\[\Rightarrow 48 = x^2 + 2x - 15\]
\[ \Rightarrow x^2 + 2x - 15 - 48 = 0\]
\[ \Rightarrow x^2 + 2x - 63 = 0\]
\[ \Rightarrow x^2 + 9x - 7x - 63 = 0\]
\[\Rightarrow x\left( x + 9 \right) - 7\left( x + 9 \right) = 0\]
\[ \Rightarrow \left( x - 7 \right)\left( x + 9 \right) = 0\]
\[ \Rightarrow x = 7, - 9\]
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