Question
Subtract and simplify. \( \frac{3}{x-3}-\frac{9}{x+3} \)
Ask by Beck Carrillo. in the United States
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified expression is \( \frac{-6x+36}{x^{2}-9} \).
Solution
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The Deep Dive
To subtract the fractions \( \frac{3}{x-3} \) and \( \frac{9}{x+3} \), first find a common denominator, which is \( (x-3)(x+3) \). Rewrite the fractions: \[ \frac{3(x+3)}{(x-3)(x+3)} - \frac{9(x-3)}{(x-3)(x+3)} \] Now, expand the numerators: \[ \frac{3x + 9 - (9x - 27)}{(x-3)(x+3)} = \frac{3x + 9 - 9x + 27}{(x-3)(x+3)} \] Combine like terms in the numerator: \[ \frac{-6x + 36}{(x-3)(x+3)} \] Factor out the numerator: \[ \frac{-6(x - 6)}{(x-3)(x+3)} \] So, the simplified expression is: \[ \frac{-6(x - 6)}{(x-3)(x+3)} \]
