Precalculus Examples

$\frac{5}{\frac{5}{x - 7} - 7}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{5}{\frac{5}{x - 7} - 7 \cdot \frac{x - 7}{x - 7}}$

Step 1.2

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

$\frac{5}{\frac{5}{x - 7} + \frac{- 7 (x - 7)}{x - 7}}$

Step 1.3

Combine the numerators over the common denominator.

$\frac{5}{\frac{5 - 7 (x - 7)}{x - 7}}$

Step 1.4

Rewrite $\frac{5 - 7 (x - 7)}{x - 7}$ in a factored form.

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Step 1.4.1

Apply the distributive property.

$\frac{5}{\frac{5 - 7 x - 7 \cdot - 7}{x - 7}}$

Step 1.4.2

Multiply $- 7$ by $- 7$ .

$\frac{5}{\frac{5 - 7 x + 49}{x - 7}}$

Step 1.4.3

Add $5$ and $49$ .

$\frac{5}{\frac{- 7 x + 54}{x - 7}}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$5 \frac{x - 7}{- 7 x + 54}$

Step 3

Combine $5$ and $\frac{x - 7}{- 7 x + 54}$ .

$\frac{5 (x - 7)}{- 7 x + 54}$

Step 4

Factor $- 1$ out of $- 7 x$ .

$\frac{5 (x - 7)}{- (7 x) + 54}$

Step 5

Rewrite $54$ as $- 1 (- 54)$ .

$\frac{5 (x - 7)}{- (7 x) - 1 (- 54)}$

Step 6

Factor $- 1$ out of $- (7 x) - 1 (- 54)$ .

$\frac{5 (x - 7)}{- (7 x - 54)}$

Step 7

Rewrite negatives.

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Step 7.1

Rewrite $- (7 x - 54)$ as $- 1 (7 x - 54)$ .

$\frac{5 (x - 7)}{- 1 (7 x - 54)}$

Step 7.2

Move the negative in front of the fraction.

$- \frac{5 (x - 7)}{7 x - 54}$