Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x - 1) (x - 3)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x - 1$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 3$ .

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Step 3.2.1

Factor $x - 3$ out of $(x - 1) (x - 3)$ .

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $x - 1$ .

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Step 3.3.1

Move the leading negative in $- \frac{3}{x - 1}$ into the numerator.

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $x - 3$ .

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Step 3.4.1

Move the leading negative in $- \frac{3}{x - 3}$ into the numerator.

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

Step 3.4.2

Factor $x - 3$ out of $(x - 1) (x - 3)$ .

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

Step 3.4.3

Cancel the common factor.

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

Step 3.4.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Factor $2$ out of $(x - 3) \cdot 2 + (x - 1) \cdot 2$ .

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Step 4.1.1

Factor $2$ out of $(x - 3) \cdot 2$ .

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

Step 4.1.2

Factor $2$ out of $(x - 1) \cdot 2$ .

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

Step 4.1.3

Factor $2$ out of $2 \cdot (x - 3) + 2 \cdot (x - 1)$ .

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

Step 4.2

Add $x$ and $x$ .

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

Step 4.3

Subtract $1$ from $- 3$ .

$\frac{2 (2 x - 4)}{(x - 3) \cdot - 3 + (x - 1) \cdot - 3}$

Step 4.4

Factor $2$ out of $2 x - 4$ .

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Step 4.4.1

Factor $2$ out of $2 x$ .

$\frac{2 (2 (x) - 4)}{(x - 3) \cdot - 3 + (x - 1) \cdot - 3}$

Step 4.4.2

Factor $2$ out of $- 4$ .

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

Step 4.4.3

Factor $2$ out of $2 x + 2 \cdot - 2$ .

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

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

Step 4.5

Multiply $2$ by $2$ .

$\frac{4 (x - 2)}{(x - 3) \cdot - 3 + (x - 1) \cdot - 3}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $3$ out of $(x - 3) \cdot - 3 + (x - 1) \cdot - 3$ .

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Step 5.1.1

Factor $3$ out of $(x - 3) \cdot - 3$ .

$\frac{4 (x - 2)}{3 ((x - 3) \cdot - 1) + (x - 1) \cdot - 3}$

Step 5.1.2

Factor $3$ out of $(x - 1) \cdot - 3$ .

$\frac{4 (x - 2)}{3 ((x - 3) \cdot - 1) + 3 ((x - 1) \cdot - 1)}$

Step 5.1.3

Factor $3$ out of $3 ((x - 3) \cdot - 1) + 3 ((x - 1) \cdot - 1)$ .

$\frac{4 (x - 2)}{3 ((x - 3) \cdot - 1 + (x - 1) \cdot - 1)}$

Step 5.2

Apply the distributive property.

$\frac{4 (x - 2)}{3 (x \cdot - 1 - 3 \cdot - 1 + (x - 1) \cdot - 1)}$

Step 5.3

Move $- 1$ to the left of $x$ .

$\frac{4 (x - 2)}{3 (- 1 \cdot x - 3 \cdot - 1 + (x - 1) \cdot - 1)}$

Step 5.4

Multiply $- 3$ by $- 1$ .

$\frac{4 (x - 2)}{3 (- 1 \cdot x + 3 + (x - 1) \cdot - 1)}$

Step 5.5

Rewrite $- 1 x$ as $- x$ .

$\frac{4 (x - 2)}{3 (- x + 3 + (x - 1) \cdot - 1)}$

Step 5.6

Apply the distributive property.

$\frac{4 (x - 2)}{3 (- x + 3 + x \cdot - 1 - 1 \cdot - 1)}$

Step 5.7

Move $- 1$ to the left of $x$ .

$\frac{4 (x - 2)}{3 (- x + 3 - 1 \cdot x - 1 \cdot - 1)}$

Step 5.8

Multiply $- 1$ by $- 1$ .

$\frac{4 (x - 2)}{3 (- x + 3 - 1 \cdot x + 1)}$

Step 5.9

Rewrite $- 1 x$ as $- x$ .

$\frac{4 (x - 2)}{3 (- x + 3 - x + 1)}$

Step 5.10

Subtract $x$ from $- x$ .

$\frac{4 (x - 2)}{3 (- 2 x + 3 + 1)}$

Step 5.11

Add $3$ and $1$ .

$\frac{4 (x - 2)}{3 (- 2 x + 4)}$

Step 5.12

Factor $2$ out of $- 2 x + 4$ .

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Step 5.12.1

Factor $2$ out of $- 2 x$ .

$\frac{4 (x - 2)}{3 (2 (- x) + 4)}$

Step 5.12.2

Factor $2$ out of $4$ .

$\frac{4 (x - 2)}{3 (2 (- x) + 2 (2))}$

Step 5.12.3

Factor $2$ out of $2 (- x) + 2 (2)$ .

$\frac{4 (x - 2)}{3 (2 (- x + 2))}$

$\frac{4 (x - 2)}{3 \cdot 2 (- x + 2)}$

Step 5.13

Multiply $3$ by $2$ .

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

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Cancel the common factor of $4$ and $6$ .

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Step 6.1.1

Factor $2$ out of $4 (x - 2)$ .

$\frac{2 (2 (x - 2))}{6 (- x + 2)}$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $6 (- x + 2)$ .

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

Step 6.1.2.2

Cancel the common factor.

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

Step 6.1.2.3

Rewrite the expression.

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

Step 6.2

Cancel the common factor of $x - 2$ and $- x + 2$ .

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Step 6.2.1

Factor $- 1$ out of $x$ .

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

Step 6.2.2

Rewrite $- 2$ as $- 1 (2)$ .

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

Step 6.2.3

Factor $- 1$ out of $- 1 (- x) - 1 (2)$ .

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

Step 6.2.4

Cancel the common factor.

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

Step 6.2.5

Rewrite the expression.

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

Step 7

Multiply $2$ by $- 1$ .

$\frac{- 2}{3}$

Step 8

Move the negative in front of the fraction.

$- \frac{2}{3}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$- \frac{2}{3}$

Decimal Form:

$- 0.\overline{6}$