Algebra Examples

$\frac{1}{x - 4} - \frac{3 x}{x^{2} - 16} = \frac{2}{x + 4}$

Step 1

Subtract $\frac{2}{x + 4}$ from both sides of the equation.

$\frac{1}{x - 4} - \frac{3 x}{x^{2} - 16} - \frac{2}{x + 4} = 0$

Step 2

Simplify $\frac{1}{x - 4} - \frac{3 x}{x^{2} - 16} - \frac{2}{x + 4}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

Rewrite $16$ as $4^{2}$ .

$\frac{1}{x - 4} - \frac{3 x}{x^{2} - 4^{2}} - \frac{2}{x + 4} = 0$

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 $b = 4$ .

$\frac{1}{x - 4} - \frac{3 x}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.2

To write $\frac{1}{x - 4}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{1}{x - 4} \cdot \frac{x + 4}{x + 4} - \frac{3 x}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.3

Write each expression with a common denominator of $(x - 4) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{x - 4}$ by $\frac{x + 4}{x + 4}$ .

$\frac{x + 4}{(x - 4) (x + 4)} - \frac{3 x}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.3.2

Reorder the factors of $(x - 4) (x + 4)$ .

$\frac{x + 4}{(x + 4) (x - 4)} - \frac{3 x}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.4

Combine the numerators over the common denominator.

$\frac{x + 4 - 3 x}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.5

Subtract $3 x$ from $x$ .

$\frac{- 2 x + 4}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.6

Factor $2$ out of $- 2 x + 4$ .

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Step 2.6.1

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

$\frac{2 (- x) + 4}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.6.2

Factor $2$ out of $4$ .

$\frac{2 (- x) + 2 (2)}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.6.3

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

$\frac{2 (- x + 2)}{(x + 4) (x - 4)} - \frac{2}{x + 4} = 0$

Step 2.7

To write $- \frac{2}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{2 (- x + 2)}{(x + 4) (x - 4)} - \frac{2}{x + 4} \cdot \frac{x - 4}{x - 4} = 0$

Step 2.8

Multiply $\frac{2}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{2 (- x + 2)}{(x + 4) (x - 4)} - \frac{2 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.9

Combine the numerators over the common denominator.

$\frac{2 (- x + 2) - 2 (x - 4)}{(x + 4) (x - 4)} = 0$

Step 2.10

Simplify the numerator.

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Step 2.10.1

Factor $2$ out of $2 (- x + 2) - 2 (x - 4)$ .

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Step 2.10.1.1

Factor $2$ out of $- 2 (x - 4)$ .

$\frac{2 (- x + 2) + 2 (- (x - 4))}{(x + 4) (x - 4)} = 0$

Step 2.10.1.2

Factor $2$ out of $2 (- x + 2) + 2 (- (x - 4))$ .

$\frac{2 (- x + 2 - (x - 4))}{(x + 4) (x - 4)} = 0$

Step 2.10.2

Apply the distributive property.

$\frac{2 (- x + 2 - x - - 4)}{(x + 4) (x - 4)} = 0$

Step 2.10.3

Multiply $- 1$ by $- 4$ .

$\frac{2 (- x + 2 - x + 4)}{(x + 4) (x - 4)} = 0$

Step 2.10.4

Subtract $x$ from $- x$ .

$\frac{2 (- 2 x + 2 + 4)}{(x + 4) (x - 4)} = 0$

Step 2.10.5

Add $2$ and $4$ .

$\frac{2 (- 2 x + 6)}{(x + 4) (x - 4)} = 0$

Step 2.10.6

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

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Step 2.10.6.1

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

$\frac{2 (2 (- x) + 6)}{(x + 4) (x - 4)} = 0$

Step 2.10.6.2

Factor $2$ out of $6$ .

$\frac{2 (2 (- x) + 2 (3))}{(x + 4) (x - 4)} = 0$

Step 2.10.6.3

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

$\frac{2 (2 (- x + 3))}{(x + 4) (x - 4)} = 0$

$\frac{2 \cdot 2 (- x + 3)}{(x + 4) (x - 4)} = 0$

Step 2.10.7

Multiply $2$ by $2$ .

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

Step 2.11

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

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

Step 2.12

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

$\frac{4 (- (x) - 1 (- 3))}{(x + 4) (x - 4)} = 0$

Step 2.13

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

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

Step 2.14

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

$\frac{4 (- 1 (x - 3))}{(x + 4) (x - 4)} = 0$

Step 2.15

Move the negative in front of the fraction.

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

Step 3

Set the numerator equal to zero.

$4 (x - 3) = 0$

Step 4

Solve the equation for $x$ .

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Step 4.1

Divide each term in $4 (x - 3) = 0$ by $4$ and simplify.

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Step 4.1.1

Divide each term in $4 (x - 3) = 0$ by $4$ .

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

Step 4.1.2

Simplify the left side.

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Step 4.1.2.1

Cancel the common factor of $4$ .

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Step 4.1.2.1.1

Cancel the common factor.

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

Step 4.1.2.1.2

Divide $x - 3$ by $1$ .

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

Step 4.1.3

Simplify the right side.

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Step 4.1.3.1

Divide $0$ by $4$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$