Algebra Examples

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

Step 1

Factor $x^{2} + 3 x - 4$ using the AC method.

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Step 1.1

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 $- 4$ and whose sum is $3$ .

$- 1, 4$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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

Step 4

Write each expression with a common denominator of $(x + 4) (x - 1)$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

Reorder the factors of $(x - 1) (x + 4)$ .

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

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

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

Step 5.2

Combine the numerators over the common denominator.

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

Step 6

Simplify each term.

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Step 6.1

Expand $(x - 4) (x - 1)$ using the FOIL Method.

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Step 6.1.1

Apply the distributive property.

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

Step 6.1.2

Apply the distributive property.

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

Step 6.1.3

Apply the distributive property.

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

Step 6.2

Simplify and combine like terms.

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Step 6.2.1

Simplify each term.

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Step 6.2.1.1

Multiply $x$ by $x$ .

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

Step 6.2.1.2

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

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

Step 6.2.1.3

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

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

Step 6.2.1.4

Multiply $- 4$ by $- 1$ .

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

Step 6.2.2

Subtract $4 x$ from $- x$ .

$\frac{x^{2} - 5 x + 4 - (x + 1) (x + 4) + x}{(x + 4) (x - 1)}$

Step 6.3

Apply the distributive property.

$\frac{x^{2} - 5 x + 4 + (- x - 1 \cdot 1) (x + 4) + x}{(x + 4) (x - 1)}$

Step 6.4

Multiply $- 1$ by $1$ .

$\frac{x^{2} - 5 x + 4 + (- x - 1) (x + 4) + x}{(x + 4) (x - 1)}$

Step 6.5

Expand $(- x - 1) (x + 4)$ using the FOIL Method.

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Step 6.5.1

Apply the distributive property.

$\frac{x^{2} - 5 x + 4 - x (x + 4) - 1 (x + 4) + x}{(x + 4) (x - 1)}$

Step 6.5.2

Apply the distributive property.

$\frac{x^{2} - 5 x + 4 - x \cdot x - x \cdot 4 - 1 (x + 4) + x}{(x + 4) (x - 1)}$

Step 6.5.3

Apply the distributive property.

$\frac{x^{2} - 5 x + 4 - x \cdot x - x \cdot 4 - 1 x - 1 \cdot 4 + x}{(x + 4) (x - 1)}$

Step 6.6

Simplify and combine like terms.

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Step 6.6.1

Simplify each term.

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Step 6.6.1.1

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

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Step 6.6.1.1.1

Move $x$ .

$\frac{x^{2} - 5 x + 4 - (x \cdot x) - x \cdot 4 - 1 x - 1 \cdot 4 + x}{(x + 4) (x - 1)}$

Step 6.6.1.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} - 5 x + 4 - x^{2} - x \cdot 4 - 1 x - 1 \cdot 4 + x}{(x + 4) (x - 1)}$

Step 6.6.1.2

Multiply $4$ by $- 1$ .

$\frac{x^{2} - 5 x + 4 - x^{2} - 4 x - 1 x - 1 \cdot 4 + x}{(x + 4) (x - 1)}$

Step 6.6.1.3

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

$\frac{x^{2} - 5 x + 4 - x^{2} - 4 x - x - 1 \cdot 4 + x}{(x + 4) (x - 1)}$

Step 6.6.1.4

Multiply $- 1$ by $4$ .

$\frac{x^{2} - 5 x + 4 - x^{2} - 4 x - x - 4 + x}{(x + 4) (x - 1)}$

Step 6.6.2

Subtract $x$ from $- 4 x$ .

$\frac{x^{2} - 5 x + 4 - x^{2} - 5 x - 4 + x}{(x + 4) (x - 1)}$

Step 7

Simplify by adding terms.

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Step 7.1

Combine the opposite terms in $x^{2} - 5 x + 4 - x^{2} - 5 x - 4 + x$ .

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Step 7.1.1

Subtract $x^{2}$ from $x^{2}$ .

$\frac{- 5 x + 4 + 0 - 5 x - 4 + x}{(x + 4) (x - 1)}$

Step 7.1.2

Add $- 5 x + 4$ and $0$ .

$\frac{- 5 x + 4 - 5 x - 4 + x}{(x + 4) (x - 1)}$

Step 7.1.3

Subtract $4$ from $4$ .

$\frac{- 5 x - 5 x + 0 + x}{(x + 4) (x - 1)}$

Step 7.1.4

Add $- 5 x - 5 x$ and $0$ .

$\frac{- 5 x - 5 x + x}{(x + 4) (x - 1)}$

Step 7.2

Subtract $5 x$ from $- 5 x$ .

$\frac{- 10 x + x}{(x + 4) (x - 1)}$

Step 7.3

Add $- 10 x$ and $x$ .

$\frac{- 9 x}{(x + 4) (x - 1)}$

Step 7.4

Move the negative in front of the fraction.

$- \frac{9 x}{(x + 4) (x - 1)}$