Pre-Algebra Examples

$\frac{- x - 2}{9 x^{2} - 1} + \frac{- 5 x + 4}{9 x^{2} - 1}$

Step 1

Simplify terms.

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Step 1.1

Combine the numerators over the common denominator.

$\frac{- x - 2 - 5 x + 4}{9 x^{2} - 1}$

Step 1.2

Subtract $5 x$ from $- x$ .

$\frac{- 6 x - 2 + 4}{9 x^{2} - 1}$

Step 1.3

Add $- 2$ and $4$ .

$\frac{- 6 x + 2}{9 x^{2} - 1}$

Step 1.4

Factor $2$ out of $- 6 x + 2$ .

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Step 1.4.1

Factor $2$ out of $- 6 x$ .

$\frac{2 (- 3 x) + 2}{9 x^{2} - 1}$

Step 1.4.2

Factor $2$ out of $2$ .

$\frac{2 (- 3 x) + 2 (1)}{9 x^{2} - 1}$

Step 1.4.3

Factor $2$ out of $2 (- 3 x) + 2 (1)$ .

$\frac{2 (- 3 x + 1)}{9 x^{2} - 1}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $9 x^{2}$ as ${(3 x)}^{2}$ .

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

Step 2.2

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

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3 x$ and $b = 1$ .

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

Step 3

Cancel the common factor of $- 3 x + 1$ and $3 x - 1$ .

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Rewrite $- (3 x - 1)$ as $- 1 (3 x - 1)$ .

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

Step 3.5

Cancel the common factor.

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

Step 3.6

Rewrite the expression.

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

Step 4

Multiply $2$ by $- 1$ .

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

Step 5

Move the negative in front of the fraction.

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