Examples

\frac{1}{x - 1} - \frac{5}{x - 6}

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

Step 1

To write

\frac{1}{x - 1}

$\frac{1}{x - 1}$ as a fraction with a common denominator, multiply by

\frac{x - 6}{x - 6}

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

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

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

Step 2

To write

- \frac{5}{x - 6}

$- \frac{5}{x - 6}$ as a fraction with a common denominator, multiply by

\frac{x - 1}{x - 1}

$\frac{x - 1}{x - 1}$ .

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

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

Step 3

Write each expression with a common denominator of

(x - 1) (x - 6)

$(x - 1) (x - 6)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{1}{x - 1}

$\frac{1}{x - 1}$ by

\frac{x - 6}{x - 6}

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

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

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

Step 3.2

Multiply

\frac{5}{x - 6}

$\frac{5}{x - 6}$ by

\frac{x - 1}{x - 1}

$\frac{x - 1}{x - 1}$ .

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

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

Step 3.3

Reorder the factors of

(x - 6) (x - 1)

$(x - 6) (x - 1)$ .

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

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

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

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

Step 4

Combine the numerators over the common denominator.

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

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

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

Step 5.2

Multiply

- 5

$- 5$ by

- 1

$- 1$ .

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

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

Step 5.3

Subtract

5 x

$5 x$ from

x

$x$ .

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

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

Step 5.4

Add

- 6

$- 6$ and

5

$5$ .

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

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

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

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

Step 6

Simplify with factoring out.

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Step 6.1

Factor

- 1

$- 1$ out of

- 4 x

$- 4 x$ .

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

Step 6.2

Rewrite

$- 1$ as

$- 1 (1)$ .

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

Step 6.3

Factor

$- 1$ out of

$- (4 x) - 1 (1)$ .

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

Step 6.4

Simplify the expression.

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Step 6.4.1

Rewrite

$- (4 x + 1)$ as

$- 1 (4 x + 1)$ .

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

Step 6.4.2

Move the negative in front of the fraction.

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

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