Simplify the expression \( \frac{\frac{6}{x-1}-\frac{1}{x+1}}{\frac{x}{x-1}+\frac{1}{x+1}} \)
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To simplify the expression \[ \frac{\frac{6}{x-1}-\frac{1}{x+1}}{\frac{x}{x-1}+\frac{1}{x+1}}, \] let's first simplify the numerator and the denominator separately. **Numerator:** The numerator is \[ \frac{6}{x-1} - \frac{1}{x+1}. \] To combine these fractions, we need a common denominator, which is \((x-1)(x+1)\): \[ \frac{6(x+1) - 1(x-1)}{(x-1)(x+1)} = \frac{6x + 6 - x + 1}{(x-1)(x+1)} = \frac{5x + 7}{(x-1)(x+1)}. \] **Denominator:** The denominator is \[ \frac{x}{x-1} + \frac{1}{x+1}. \] Again, we use the common denominator \((x-1)(x+1)\): \[ \frac{x(x+1) + 1(x-1)}{(x-1)(x+1)} = \frac{x^2 + x + x - 1}{(x-1)(x+1)} = \frac{x^2 + 2x - 1}{(x-1)(x+1)}. \] **Putting it all together:** We substitute back into our original expression: \[ \frac{\frac{5x+7}{(x-1)(x+1)}}{\frac{x^2 + 2x - 1}{(x-1)(x+1)}}. \] Since the denominators are the same, we can simplify to: \[ \frac{5x + 7}{x^2 + 2x - 1}. \] Thus, the simplified expression is \[ \frac{5x + 7}{x^2 + 2x - 1}. \]
