Pre-Algebra Examples

$\frac{7 x}{x + 1} + \frac{8}{x - 1} - \frac{14}{x^{2} - 1}$

Step 1

Simplify the denominator.

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Step 1.1

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

$\frac{7 x}{x + 1} + \frac{8}{x - 1} - \frac{14}{x^{2} - 1^{2}}$

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 $b = 1$ .

$\frac{7 x}{x + 1} + \frac{8}{x - 1} - \frac{14}{(x + 1) (x - 1)}$

Step 2

To write $\frac{7 x}{x + 1}$ as a fraction with a common denominator, multiply by $\frac{x - 1}{x - 1}$ .

$\frac{7 x}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{8}{x - 1} - \frac{14}{(x + 1) (x - 1)}$

Step 3

To write $\frac{8}{x - 1}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{7 x}{x + 1} \cdot \frac{x - 1}{x - 1} + \frac{8}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{14}{(x + 1) (x - 1)}$

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 $\frac{7 x}{x + 1}$ by $\frac{x - 1}{x - 1}$ .

$\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{8}{x - 1} \cdot \frac{x + 1}{x + 1} - \frac{14}{(x + 1) (x - 1)}$

Step 4.2

Multiply $\frac{8}{x - 1}$ by $\frac{x + 1}{x + 1}$ .

$\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{8 (x + 1)}{(x - 1) (x + 1)} - \frac{14}{(x + 1) (x - 1)}$

Step 4.3

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

$\frac{7 x (x - 1)}{(x + 1) (x - 1)} + \frac{8 (x + 1)}{(x + 1) (x - 1)} - \frac{14}{(x + 1) (x - 1)}$

Step 5

Simplify terms.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{7 x (x - 1) + 8 (x + 1)}{(x + 1) (x - 1)} - \frac{14}{(x + 1) (x - 1)}$

Step 5.2

Combine the numerators over the common denominator.

$\frac{7 x (x - 1) + 8 (x + 1) - 14}{(x + 1) (x - 1)}$

Step 6

Simplify each term.

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Step 6.1

Apply the distributive property.

$\frac{7 x \cdot x + 7 x \cdot - 1 + 8 (x + 1) - 14}{(x + 1) (x - 1)}$

Step 6.2

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

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Step 6.2.1

Move $x$ .

$\frac{7 (x \cdot x) + 7 x \cdot - 1 + 8 (x + 1) - 14}{(x + 1) (x - 1)}$

Step 6.2.2

Multiply $x$ by $x$ .

$\frac{7 x^{2} + 7 x \cdot - 1 + 8 (x + 1) - 14}{(x + 1) (x - 1)}$

Step 6.3

Multiply $- 1$ by $7$ .

$\frac{7 x^{2} - 7 x + 8 (x + 1) - 14}{(x + 1) (x - 1)}$

Step 6.4

Apply the distributive property.

$\frac{7 x^{2} - 7 x + 8 x + 8 \cdot 1 - 14}{(x + 1) (x - 1)}$

Step 6.5

Multiply $8$ by $1$ .

$\frac{7 x^{2} - 7 x + 8 x + 8 - 14}{(x + 1) (x - 1)}$

Step 7

Simplify by adding terms.

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Step 7.1

Add $- 7 x$ and $8 x$ .

$\frac{7 x^{2} + x + 8 - 14}{(x + 1) (x - 1)}$

Step 7.2

Subtract $14$ from $8$ .

$\frac{7 x^{2} + x - 6}{(x + 1) (x - 1)}$

Step 8

Factor by grouping.

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Step 8.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 7 \cdot - 6 = - 42$ and whose sum is $b = 1$ .

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Step 8.1.1

Multiply by $1$ .

$\frac{7 x^{2} + 1 x - 6}{(x + 1) (x - 1)}$

Step 8.1.2

Rewrite $1$ as $- 6$ plus $7$

$\frac{7 x^{2} + (- 6 + 7) x - 6}{(x + 1) (x - 1)}$

Step 8.1.3

Apply the distributive property.

$\frac{7 x^{2} - 6 x + 7 x - 6}{(x + 1) (x - 1)}$

Step 8.2

Factor out the greatest common factor from each group.

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Step 8.2.1

Group the first two terms and the last two terms.

$\frac{(7 x^{2} - 6 x) + 7 x - 6}{(x + 1) (x - 1)}$

Step 8.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (7 x - 6) + 1 (7 x - 6)}{(x + 1) (x - 1)}$

Step 8.3

Factor the polynomial by factoring out the greatest common factor, $7 x - 6$ .

$\frac{(7 x - 6) (x + 1)}{(x + 1) (x - 1)}$

Step 9

Cancel the common factor of $x + 1$ .

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Step 9.1

Cancel the common factor.

$\frac{(7 x - 6) (x + 1)}{(x + 1) (x - 1)}$

Step 9.2

Rewrite the expression.

$\frac{7 x - 6}{x - 1}$