Finite Math Examples

$\frac{6}{x - 6} - \frac{7}{x + 6} = \frac{6}{x^{2} - 36}$

Step 1

Subtract $\frac{6}{x^{2} - 36}$ from both sides of the equation.

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{x^{2} - 36} = 0$

Step 2

Simplify $\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{x^{2} - 36}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $36$ as $6^{2}$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{x^{2} - 6^{2}} = 0$

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 $b = 6$ .

$\frac{6}{x - 6} - \frac{7}{x + 6} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.2

To write $\frac{6}{x - 6}$ as a fraction with a common denominator, multiply by $\frac{x + 6}{x + 6}$ .

$\frac{6}{x - 6} \cdot \frac{x + 6}{x + 6} - \frac{7}{x + 6} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.3

To write $- \frac{7}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x - 6}{x - 6}$ .

$\frac{6}{x - 6} \cdot \frac{x + 6}{x + 6} - \frac{7}{x + 6} \cdot \frac{x - 6}{x - 6} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.4

Write each expression with a common denominator of $(x - 6) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{6}{x - 6}$ by $\frac{x + 6}{x + 6}$ .

$\frac{6 (x + 6)}{(x - 6) (x + 6)} - \frac{7}{x + 6} \cdot \frac{x - 6}{x - 6} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.4.2

Multiply $\frac{7}{x + 6}$ by $\frac{x - 6}{x - 6}$ .

$\frac{6 (x + 6)}{(x - 6) (x + 6)} - \frac{7 (x - 6)}{(x + 6) (x - 6)} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.4.3

Reorder the factors of $(x - 6) (x + 6)$ .

$\frac{6 (x + 6)}{(x + 6) (x - 6)} - \frac{7 (x - 6)}{(x + 6) (x - 6)} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{6 (x + 6) - 7 (x - 6)}{(x + 6) (x - 6)} - \frac{6}{(x + 6) (x - 6)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{6 (x + 6) - 7 (x - 6) - 6}{(x + 6) (x - 6)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{6 x + 6 \cdot 6 - 7 (x - 6) - 6}{(x + 6) (x - 6)} = 0$

Step 2.7.2

Multiply $6$ by $6$ .

$\frac{6 x + 36 - 7 (x - 6) - 6}{(x + 6) (x - 6)} = 0$

Step 2.7.3

Apply the distributive property.

$\frac{6 x + 36 - 7 x - 7 \cdot - 6 - 6}{(x + 6) (x - 6)} = 0$

Step 2.7.4

Multiply $- 7$ by $- 6$ .

$\frac{6 x + 36 - 7 x + 42 - 6}{(x + 6) (x - 6)} = 0$

Step 2.8

Subtract $7 x$ from $6 x$ .

$\frac{- x + 36 + 42 - 6}{(x + 6) (x - 6)} = 0$

Step 2.9

Add $36$ and $42$ .

$\frac{- x + 78 - 6}{(x + 6) (x - 6)} = 0$

Step 2.10

Subtract $6$ from $78$ .

$\frac{- x + 72}{(x + 6) (x - 6)} = 0$

Step 2.11

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

$\frac{- (x) + 72}{(x + 6) (x - 6)} = 0$

Step 2.12

Rewrite $72$ as $- 1 (- 72)$ .

$\frac{- (x) - 1 (- 72)}{(x + 6) (x - 6)} = 0$

Step 2.13

Factor $- 1$ out of $- (x) - 1 (- 72)$ .

$\frac{- (x - 72)}{(x + 6) (x - 6)} = 0$

Step 2.14

Rewrite $- (x - 72)$ as $- 1 (x - 72)$ .

$\frac{- 1 (x - 72)}{(x + 6) (x - 6)} = 0$

Step 2.15

Move the negative in front of the fraction.

$- \frac{x - 72}{(x + 6) (x - 6)} = 0$

Step 3

Set the numerator equal to zero.

$x - 72 = 0$

Step 4

Add $72$ to both sides of the equation.

$x = 72$