Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

Simplify with factoring out.

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Step 2.1

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

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

Step 2.2

Factor $- 1$ out of $- x$ .

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

Step 2.3

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

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

Step 2.4

Simplify the expression.

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Step 2.4.1

Move a negative from the denominator of $\frac{3 x}{- 1 (- 1 + x)}$ to the numerator.

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

Step 2.4.2

Reorder terms.

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

Step 3

To write $\frac{- (3 x)}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Reorder the factors of $(x - 1) (x + 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $4 x - 1 \cdot 3 x (x + 1)$ .

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Step 6.1.1

Factor $x$ out of $4 x$ .

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

Step 6.1.2

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

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

Step 6.1.3

Factor $x$ out of $x \cdot 4 + x (- 1 \cdot 3 (x + 1))$ .

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

Step 6.2

Multiply $- 1$ by $3$ .

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $- 3$ by $1$ .

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

Step 6.5

Subtract $3$ from $4$ .

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

Step 7

To write $- \frac{4}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

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

Step 8

Simplify terms.

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Step 8.1

Multiply $\frac{4}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

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

Step 8.2

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Apply the distributive property.

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

Step 9.2

Rewrite using the commutative property of multiplication.

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

Step 9.3

Multiply $x$ by $1$ .

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

Step 9.4

Multiply $x$ by $x$ by adding the exponents.

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Step 9.4.1

Move $x$ .

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

Step 9.4.2

Multiply $x$ by $x$ .

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

Step 9.5

Apply the distributive property.

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

Step 9.6

Multiply $- 4$ by $- 1$ .

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

Step 9.7

Subtract $4 x$ from $x$ .

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

Step 10

Simplify with factoring out.

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Step 10.1

Factor $- 1$ out of $- 3 x^{2}$ .

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

Step 10.2

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

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

Step 10.3

Factor $- 1$ out of $- (3 x^{2}) - (3 x)$ .

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

Step 10.4

Rewrite $4$ as $- 1 (- 4)$ .

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

Step 10.5

Factor $- 1$ out of $- (3 x^{2} + 3 x) - 1 (- 4)$ .

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

Step 10.6

Simplify the expression.

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Step 10.6.1

Rewrite $- (3 x^{2} + 3 x - 4)$ as $- 1 (3 x^{2} + 3 x - 4)$ .

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

Step 10.6.2

Move the negative in front of the fraction.

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