Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

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

Step 2

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

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Expand $(x + 2) (x - 1)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $x$ by $x$ .

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

Step 4.2.1.2

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

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

Step 4.2.1.3

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

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

Step 4.2.1.4

Multiply $2$ by $- 1$ .

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

Step 4.2.2

Add $- x$ and $2 x$ .

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

Step 4.3

Subtract $2 x$ from $x$ .

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

Step 4.4

Factor $x^{2} - x - 2$ using the AC method.

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Step 4.4.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $- 1$ .

$- 2, 1$

Step 4.4.2

Write the factored form using these integers.

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

Step 5

Cancel the common factor of $x + 1$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Rewrite the expression.

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