Simplify (x/(x-7))/(x/(x-7)-7)

$\frac{\frac{x}{x - 7}}{\frac{x}{x - 7} - 7}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x}{x - 7} - 7}$

Step 2

Simplify the denominator.

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x}{x - 7} - 7 \cdot \frac{x - 7}{x - 7}}$

Combine $- 7$ and $\frac{x - 7}{x - 7}$ .

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x}{x - 7} + \frac{- 7 (x - 7)}{x - 7}}$

Combine the numerators over the common denominator.

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x - 7 (x - 7)}{x - 7}}$

Rewrite $\frac{x - 7 (x - 7)}{x - 7}$ in a factored form.

Apply the distributive property.

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x - 7 x - 7 \cdot - 7}{x - 7}}$

Multiply $- 7$ by $- 7$ .

$\frac{x}{x - 7} \cdot \frac{1}{\frac{x - 7 x + 49}{x - 7}}$

Subtract $7 x$ from $x$ .

$\frac{x}{x - 7} \cdot \frac{1}{\frac{- 6 x + 49}{x - 7}}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$\frac{x}{x - 7} (1 \frac{x - 7}{- 6 x + 49})$

Step 4

Multiply $\frac{x - 7}{- 6 x + 49}$ by $1$ .

$\frac{x}{x - 7} \cdot \frac{x - 7}{- 6 x + 49}$

Step 5

Cancel the common factor of $x - 7$ .

Cancel the common factor.

$\frac{x}{x - 7} \cdot \frac{x - 7}{- 6 x + 49}$

Rewrite the expression.

$x \frac{1}{- 6 x + 49}$

Step 6

Combine $x$ and $\frac{1}{- 6 x + 49}$ .

$\frac{x}{- 6 x + 49}$

Step 7

Factor $- 1$ out of $- 6 x$ .

$\frac{x}{- (6 x) + 49}$

Step 8

Rewrite $49$ as $- 1 (- 49)$ .

$\frac{x}{- (6 x) - 1 (- 49)}$

Step 9

Factor $- 1$ out of $- (6 x) - 1 (- 49)$ .

$\frac{x}{- (6 x - 49)}$

Step 10

Rewrite negatives.

Rewrite $- (6 x - 49)$ as $- 1 (6 x - 49)$ .

$\frac{x}{- 1 (6 x - 49)}$

Move the negative in front of the fraction.

$- \frac{x}{6 x - 49}$

Simplify (x/(x-7))/(x/(x-7)-7)

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