Algebra Examples

\frac{2}{x - 3} - \frac{1}{x - 1}

$\frac{2}{x - 3} - \frac{1}{x - 1}$

Step 1

To write

\frac{2}{x - 3}

$\frac{2}{x - 3}$ as a fraction with a common denominator, multiply by

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

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

\frac{2}{x - 3} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1}

$\frac{2}{x - 3} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1}$

Step 2

To write

- \frac{1}{x - 1}

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

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

\frac{2}{x - 3} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{x - 3}{x - 3}

$\frac{2}{x - 3} \cdot \frac{x - 1}{x - 1} - \frac{1}{x - 1} \cdot \frac{x - 3}{x - 3}$

Step 3

Write each expression with a common denominator of

(x - 3) (x - 1)

$(x - 3) (x - 1)$ , by multiplying each by an appropriate factor of

1

$1$ .

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Step 3.1

Multiply

\frac{2}{x - 3}

$\frac{2}{x - 3}$ by

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

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

\frac{2 (x - 1)}{(x - 3) (x - 1)} - \frac{1}{x - 1} \cdot \frac{x - 3}{x - 3}

$\frac{2 (x - 1)}{(x - 3) (x - 1)} - \frac{1}{x - 1} \cdot \frac{x - 3}{x - 3}$

Step 3.2

Multiply

\frac{1}{x - 1}

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

\frac{x - 3}{x - 3}

$\frac{x - 3}{x - 3}$ .

\frac{2 (x - 1)}{(x - 3) (x - 1)} - \frac{x - 3}{(x - 1) (x - 3)}

$\frac{2 (x - 1)}{(x - 3) (x - 1)} - \frac{x - 3}{(x - 1) (x - 3)}$

Step 3.3

Reorder the factors of

(x - 3) (x - 1)

$(x - 3) (x - 1)$ .

\frac{2 (x - 1)}{(x - 1) (x - 3)} - \frac{x - 3}{(x - 1) (x - 3)}

$\frac{2 (x - 1)}{(x - 1) (x - 3)} - \frac{x - 3}{(x - 1) (x - 3)}$

\frac{2 (x - 1)}{(x - 1) (x - 3)} - \frac{x - 3}{(x - 1) (x - 3)}

$\frac{2 (x - 1)}{(x - 1) (x - 3)} - \frac{x - 3}{(x - 1) (x - 3)}$

Step 4

Combine the numerators over the common denominator.

\frac{2 (x - 1) - (x - 3)}{(x - 1) (x - 3)}

$\frac{2 (x - 1) - (x - 3)}{(x - 1) (x - 3)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

\frac{2 x + 2 \cdot - 1 - (x - 3)}{(x - 1) (x - 3)}

$\frac{2 x + 2 \cdot - 1 - (x - 3)}{(x - 1) (x - 3)}$

Step 5.2

Multiply

2

$2$ by

- 1

$- 1$ .

\frac{2 x - 2 - (x - 3)}{(x - 1) (x - 3)}

$\frac{2 x - 2 - (x - 3)}{(x - 1) (x - 3)}$

Step 5.3

Apply the distributive property.

\frac{2 x - 2 - x - - 3}{(x - 1) (x - 3)}

$\frac{2 x - 2 - x - - 3}{(x - 1) (x - 3)}$

Step 5.4

Multiply

- 1

$- 1$ by

- 3

$- 3$ .

\frac{2 x - 2 - x + 3}{(x - 1) (x - 3)}

$\frac{2 x - 2 - x + 3}{(x - 1) (x - 3)}$

Step 5.5

Subtract

x

$x$ from

2 x

$2 x$ .

\frac{x - 2 + 3}{(x - 1) (x - 3)}

$\frac{x - 2 + 3}{(x - 1) (x - 3)}$

Step 5.6

Add

- 2

$- 2$ and

3

$3$ .

\frac{x + 1}{(x - 1) (x - 3)}

$\frac{x + 1}{(x - 1) (x - 3)}$

\frac{x + 1}{(x - 1) (x - 3)}

$\frac{x + 1}{(x - 1) (x - 3)}$