Solve for x : 1/(x − 3) − 1/(x + 5) = 1/6
Question
Solve for x:
\[ \frac{1}{x-3}-\frac{1}{x+5}=\frac{1}{6}, \qquad x\ne3,-5 \]Solution
Taking LCM on the left side,
\[ \frac{(x+5)-(x-3)}{(x-3)(x+5)} = \frac{1}{6} \]
\[ \frac{8}{(x-3)(x+5)} = \frac{1}{6} \]
Cross multiplying,
\[ 48=(x-3)(x+5) \]
\[ 48=x^2+2x-15 \]
\[ x^2+2x-63=0 \]
\[ (x+9)(x-7)=0 \]
\[ x=-9 \]
or
\[ x=7 \]
Both values satisfy the given restrictions.
Answer
\[
\boxed{x=-9 \quad \text{or} \quad x=7}
\]