Pre-Algebra Examples

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

Step 1

Simplify the denominator.

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Step 1.1

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

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

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 $b = 1$ .

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

Step 2

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

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

Step 3

To write $- \frac{3}{(x + 1) (x - 1)}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Step 4

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

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Step 4.1

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

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

Step 4.2

Multiply $\frac{3}{(x + 1) (x - 1)}$ by $\frac{x - 2}{x - 2}$ .

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

Step 4.3

Reorder the factors of $(x + 1) (x - 2) (x - 1)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

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

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Step 6.2.1

Move $x$ .

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

Step 6.2.2

Multiply $x$ by $x$ .

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

Step 6.3

Multiply $- 1$ by $2$ .

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

Step 6.4

Apply the distributive property.

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

Step 6.5

Multiply $- 3$ by $- 2$ .

$\frac{2 x^{2} - 2 x - 3 x + 6}{(x + 1) (x - 1) (x - 2)}$

Step 6.6

Subtract $3 x$ from $- 2 x$ .

$\frac{2 x^{2} - 5 x + 6}{(x + 1) (x - 1) (x - 2)}$