Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(x + 2) (x - 2)$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Move the leading negative in $- \frac{1}{x + 2}$ into the numerator.

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x - 2$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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Step 4.1.1

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

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

Step 4.1.2

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

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

Step 4.1.3

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

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

Step 4.2

Multiply $x + 2$ by $1$ .

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

Step 4.3

Subtract $1$ from $2$ .

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

Step 5

Simplify the denominator.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Multiply $x - 2$ by $1$ .

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

Step 5.3

Add $- 2$ and $3$ .

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

Step 6

Cancel the common factor of $x + 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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