Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(2 + x) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{1}{2 + x}$ by $\frac{x}{x}$ .

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

Step 3.2

Multiply $\frac{2 - x}{x}$ by $\frac{2 + x}{2 + x}$ .

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

Step 3.3

Reorder the factors of $(2 + x) x$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $- 1$ by $2$ .

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

Step 5.3

Multiply $- - x$ .

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

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

Step 5.3.2

Multiply $x$ by $1$ .

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

Step 5.4

Expand $(- 2 + x) (2 + x)$ using the FOIL Method.

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Step 5.4.1

Apply the distributive property.

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

Step 5.4.2

Apply the distributive property.

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

Step 5.4.3

Apply the distributive property.

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

Step 5.5

Combine the opposite terms in $- 2 \cdot 2 - 2 x + x \cdot 2 + x \cdot x$ .

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Step 5.5.1

Reorder the factors in the terms $- 2 x$ and $x \cdot 2$ .

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

Step 5.5.2

Add $- 2 x$ and $2 x$ .

$\frac{x - 2 \cdot 2 + 0 + x \cdot x}{x (2 + x)}$

Step 5.5.3

Add $- 2 \cdot 2$ and $0$ .

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

Step 5.6

Simplify each term.

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Step 5.6.1

Multiply $- 2$ by $2$ .

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

Step 5.6.2

Multiply $x$ by $x$ .

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

Step 5.7

Reorder terms.

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