Algebra Examples

$\frac{6}{x^{2} - 9} - \frac{1}{x - 3} = 1$

Step 1

Subtract $1$ from both sides of the equation.

$\frac{6}{x^{2} - 9} - \frac{1}{x - 3} - 1 = 0$

Step 2

Simplify $\frac{6}{x^{2} - 9} - \frac{1}{x - 3} - 1$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $9$ as $3^{2}$ .

$\frac{6}{x^{2} - 3^{2}} - \frac{1}{x - 3} - 1 = 0$

Step 2.1.2

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 $b = 3$ .

$\frac{6}{(x + 3) (x - 3)} - \frac{1}{x - 3} - 1 = 0$

Step 2.2

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

$\frac{6}{(x + 3) (x - 3)} - \frac{1}{x - 3} \cdot \frac{x + 3}{x + 3} - 1 = 0$

Step 2.3

Write each expression with a common denominator of $(x + 3) (x - 3)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

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

$\frac{6}{(x + 3) (x - 3)} - \frac{x + 3}{(x - 3) (x + 3)} - 1 = 0$

Step 2.3.2

Reorder the factors of $(x - 3) (x + 3)$ .

$\frac{6}{(x + 3) (x - 3)} - \frac{x + 3}{(x + 3) (x - 3)} - 1 = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{6 - (x + 3)}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5

Simplify each term.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Apply the distributive property.

$\frac{6 - x - 1 \cdot 3}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.1.2

Multiply $- 1$ by $3$ .

$\frac{6 - x - 3}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.1.3

Subtract $3$ from $6$ .

$\frac{- x + 3}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2

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

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Step 2.5.2.1

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

$\frac{- (x) + 3}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2.2

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

$\frac{- (x) - 1 (- 3)}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2.3

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

$\frac{- (x - 3)}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2.4

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

$\frac{- 1 (x - 3)}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2.5

Cancel the common factor.

$\frac{- 1 (x - 3)}{(x + 3) (x - 3)} - 1 = 0$

Step 2.5.2.6

Rewrite the expression.

$\frac{- 1}{x + 3} - 1 = 0$

Step 2.5.3

Move the negative in front of the fraction.

$- \frac{1}{x + 3} - 1 = 0$

Step 2.6

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

$- \frac{1}{x + 3} - 1 \cdot \frac{x + 3}{x + 3} = 0$

Step 2.7

Combine $- 1$ and $\frac{x + 3}{x + 3}$ .

$- \frac{1}{x + 3} + \frac{- (x + 3)}{x + 3} = 0$

Step 2.8

Combine the numerators over the common denominator.

$\frac{- 1 - (x + 3)}{x + 3} = 0$

Step 2.9

Simplify the numerator.

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Step 2.9.1

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

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Step 2.9.1.1

Reorder $- 1$ and $- (x + 3)$ .

$\frac{- (x + 3) - 1}{x + 3} = 0$

Step 2.9.1.2

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

$\frac{- (x + 3) - 1 (1)}{x + 3} = 0$

Step 2.9.1.3

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

$\frac{- (x + 3 + 1)}{x + 3} = 0$

Step 2.9.2

Add $3$ and $1$ .

$\frac{- (x + 4)}{x + 3} = 0$

Step 2.10

Move the negative in front of the fraction.

$- \frac{x + 4}{x + 3} = 0$

Step 3

Set the numerator equal to zero.

$x + 4 = 0$

Step 4

Subtract $4$ from both sides of the equation.

$x = - 4$