Simplify (x/(x-5))/(x/(x-5)-5)

$\frac{\frac{x}{x - 5}}{\frac{x}{x - 5} - 5}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x}{x - 5} - 5}$

Step 2

Simplify the denominator.

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x}{x - 5} - 5 \cdot \frac{x - 5}{x - 5}}$

Combine $- 5$ and $\frac{x - 5}{x - 5}$ .

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x}{x - 5} + \frac{- 5 (x - 5)}{x - 5}}$

Combine the numerators over the common denominator.

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x - 5 (x - 5)}{x - 5}}$

Rewrite $\frac{x - 5 (x - 5)}{x - 5}$ in a factored form.

Apply the distributive property.

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x - 5 x - 5 \cdot - 5}{x - 5}}$

Multiply $- 5$ by $- 5$ .

$\frac{x}{x - 5} \cdot \frac{1}{\frac{x - 5 x + 25}{x - 5}}$

Subtract $5 x$ from $x$ .

$\frac{x}{x - 5} \cdot \frac{1}{\frac{- 4 x + 25}{x - 5}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{x - 5} (1 \frac{x - 5}{- 4 x + 25})$

Step 4

Multiply $\frac{x - 5}{- 4 x + 25}$ by $1$ .

$\frac{x}{x - 5} \cdot \frac{x - 5}{- 4 x + 25}$

Step 5

Cancel the common factor of $x - 5$ .

Cancel the common factor.

$\frac{x}{x - 5} \cdot \frac{x - 5}{- 4 x + 25}$

Rewrite the expression.

$x \frac{1}{- 4 x + 25}$

Step 6

Combine $x$ and $\frac{1}{- 4 x + 25}$ .

$\frac{x}{- 4 x + 25}$

Step 7

Factor $- 1$ out of $- 4 x$ .

$\frac{x}{- (4 x) + 25}$

Step 8

Rewrite $25$ as $- 1 (- 25)$ .

$\frac{x}{- (4 x) - 1 (- 25)}$

Step 9

Factor $- 1$ out of $- (4 x) - 1 (- 25)$ .

$\frac{x}{- (4 x - 25)}$

Step 10

Rewrite negatives.

Rewrite $- (4 x - 25)$ as $- 1 (4 x - 25)$ .

$\frac{x}{- 1 (4 x - 25)}$

Move the negative in front of the fraction.

$- \frac{x}{4 x - 25}$

Simplify (x/(x-5))/(x/(x-5)-5)

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