Questions: Subtract. 9/(x^2-9x) - x/(9x-81) 9/(x^2-9x) - x/(9x-81)=

Solution Steps

To subtract the given rational expressions, we need to follow these steps:

1. Factor the denominators to find a common denominator.
2. Rewrite each fraction with the common denominator.
3. Subtract the numerators and simplify the resulting expression.

We start with the expressions: \[ \frac{9}{x^2 - 9x} - \frac{x}{9x - 81} \] Factoring the denominators, we have: \[ x^2 - 9x = x(x - 9) \quad \text{and} \quad 9x - 81 = 9(x - 9) \]

The common denominator for the two fractions is: \[ 9(x)(x - 9) \]

Rewriting each fraction with the common denominator: \[ \frac{9 \cdot 9(x - 9)}{9(x)(x - 9)} - \frac{x \cdot (x)(x - 9)}{9(x)(x - 9)} \] This simplifies to: \[ \frac{81(x - 9) - x^2(x - 9)}{9x(x - 9)} \]

Combining the numerators: \[ 81(x - 9) - x^2(x - 9) = (81 - x^2)(x - 9) \] Thus, the expression becomes: \[ \frac{(81 - x^2)(x - 9)}{9x(x - 9)} \] Cancelling \( (x - 9) \) from the numerator and denominator (assuming \( x \neq 9 \)): \[ \frac{81 - x^2}{9x} \]

The expression \( 81 - x^2 \) can be factored as: \[ 81 - x^2 = (9 - x)(9 + x) \] Thus, the final simplified expression is: \[ \frac{(9 - x)(9 + x)}{9x} \]

Final Answer