Algebra Examples

$\frac{\frac{2}{x - 1} - \frac{7}{x - 4}}{\frac{6}{x - 1}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$(\frac{2}{x - 1} - \frac{7}{x - 4}) \frac{x - 1}{6}$

Step 2

To write $\frac{2}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$(\frac{2}{x - 1} \cdot \frac{x - 4}{x - 4} - \frac{7}{x - 4}) \frac{x - 1}{6}$

Step 3

To write $- \frac{7}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$(\frac{2}{x - 1} \cdot \frac{x - 4}{x - 4} - \frac{7}{x - 4} \cdot \frac{x - 1}{x - 1}) \frac{x - 1}{6}$

Step 4

Write each expression with a common denominator of $(x - 1) (x - 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{2}{x - 1}$ by $\frac{x - 4}{x - 4}$ .

$(\frac{2 (x - 4)}{(x - 1) (x - 4)} - \frac{7}{x - 4} \cdot \frac{x - 1}{x - 1}) \frac{x - 1}{6}$

Step 4.2

Multiply $\frac{7}{x - 4}$ by $\frac{x - 1}{x - 1}$ .

$(\frac{2 (x - 4)}{(x - 1) (x - 4)} - \frac{7 (x - 1)}{(x - 4) (x - 1)}) \frac{x - 1}{6}$

Step 4.3

Reorder the factors of $(x - 4) (x - 1)$ .

$(\frac{2 (x - 4)}{(x - 1) (x - 4)} - \frac{7 (x - 1)}{(x - 1) (x - 4)}) \frac{x - 1}{6}$

Step 5

Combine the numerators over the common denominator.

$\frac{2 (x - 4) - 7 (x - 1)}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{2 x + 2 \cdot - 4 - 7 (x - 1)}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6.2

Multiply $2$ by $- 4$ .

$\frac{2 x - 8 - 7 (x - 1)}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6.3

Apply the distributive property.

$\frac{2 x - 8 - 7 x - 7 \cdot - 1}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6.4

Multiply $- 7$ by $- 1$ .

$\frac{2 x - 8 - 7 x + 7}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6.5

Subtract $7 x$ from $2 x$ .

$\frac{- 5 x - 8 + 7}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 6.6

Add $- 8$ and $7$ .

$\frac{- 5 x - 1}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 1$ .

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Step 7.1.1

Cancel the common factor.

$\frac{- 5 x - 1}{(x - 1) (x - 4)} \cdot \frac{x - 1}{6}$

Step 7.1.2

Rewrite the expression.

$\frac{- 5 x - 1}{x - 4} \cdot \frac{1}{6}$

Step 7.2

Multiply $\frac{- 5 x - 1}{x - 4}$ by $\frac{1}{6}$ .

$\frac{- 5 x - 1}{(x - 4) \cdot 6}$

Step 7.3

Move $6$ to the left of $x - 4$ .

$\frac{- 5 x - 1}{6 \cdot (x - 4)}$

Step 7.4

Factor $- 1$ out of $- 5 x$ .

$\frac{- (5 x) - 1}{6 (x - 4)}$

Step 7.5

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

$\frac{- (5 x) - 1 (1)}{6 (x - 4)}$

Step 7.6

Factor $- 1$ out of $- (5 x) - 1 (1)$ .

$\frac{- (5 x + 1)}{6 (x - 4)}$

Step 7.7

Simplify the expression.

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Step 7.7.1

Rewrite $- (5 x + 1)$ as $- 1 (5 x + 1)$ .

$\frac{- 1 (5 x + 1)}{6 (x - 4)}$

Step 7.7.2

Move the negative in front of the fraction.

$- \frac{5 x + 1}{6 (x - 4)}$