Algebra Examples

$\frac{5}{x + 4} - \frac{2}{x - 4} = \frac{5 x}{x^{2} - 16}$

Step 1

Subtract $\frac{5 x}{x^{2} - 16}$ from both sides of the equation.

$\frac{5}{x + 4} - \frac{2}{x - 4} - \frac{5 x}{x^{2} - 16} = 0$

Step 2

Simplify $\frac{5}{x + 4} - \frac{2}{x - 4} - \frac{5 x}{x^{2} - 16}$ .

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Step 2.1

Simplify the denominator.

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Step 2.1.1

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

$\frac{5}{x + 4} - \frac{2}{x - 4} - \frac{5 x}{x^{2} - 4^{2}} = 0$

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 $b = 4$ .

$\frac{5}{x + 4} - \frac{2}{x - 4} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.2

To write $\frac{5}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{x - 4}{x - 4}$ .

$\frac{5}{x + 4} \cdot \frac{x - 4}{x - 4} - \frac{2}{x - 4} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.3

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

$\frac{5}{x + 4} \cdot \frac{x - 4}{x - 4} - \frac{2}{x - 4} \cdot \frac{x + 4}{x + 4} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.4

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 2.4.1

Multiply $\frac{5}{x + 4}$ by $\frac{x - 4}{x - 4}$ .

$\frac{5 (x - 4)}{(x + 4) (x - 4)} - \frac{2}{x - 4} \cdot \frac{x + 4}{x + 4} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.4.2

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

$\frac{5 (x - 4)}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x - 4) (x + 4)} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.4.3

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

$\frac{5 (x - 4)}{(x + 4) (x - 4)} - \frac{2 (x + 4)}{(x + 4) (x - 4)} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{5 (x - 4) - 2 (x + 4)}{(x + 4) (x - 4)} - \frac{5 x}{(x + 4) (x - 4)} = 0$

Step 2.6

Combine the numerators over the common denominator.

$\frac{5 (x - 4) - 2 (x + 4) - 5 x}{(x + 4) (x - 4)} = 0$

Step 2.7

Simplify each term.

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Step 2.7.1

Apply the distributive property.

$\frac{5 x + 5 \cdot - 4 - 2 (x + 4) - 5 x}{(x + 4) (x - 4)} = 0$

Step 2.7.2

Multiply $5$ by $- 4$ .

$\frac{5 x - 20 - 2 (x + 4) - 5 x}{(x + 4) (x - 4)} = 0$

Step 2.7.3

Apply the distributive property.

$\frac{5 x - 20 - 2 x - 2 \cdot 4 - 5 x}{(x + 4) (x - 4)} = 0$

Step 2.7.4

Multiply $- 2$ by $4$ .

$\frac{5 x - 20 - 2 x - 8 - 5 x}{(x + 4) (x - 4)} = 0$

Step 2.8

Combine the opposite terms in $5 x - 20 - 2 x - 8 - 5 x$ .

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Step 2.8.1

Subtract $5 x$ from $5 x$ .

$\frac{- 20 - 2 x - 8 + 0}{(x + 4) (x - 4)} = 0$

Step 2.8.2

Add $- 20 - 2 x - 8$ and $0$ .

$\frac{- 20 - 2 x - 8}{(x + 4) (x - 4)} = 0$

Step 2.9

Subtract $8$ from $- 20$ .

$\frac{- 2 x - 28}{(x + 4) (x - 4)} = 0$

Step 2.10

Factor $2$ out of $- 2 x - 28$ .

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Step 2.10.1

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

$\frac{2 (- x) - 28}{(x + 4) (x - 4)} = 0$

Step 2.10.2

Factor $2$ out of $- 28$ .

$\frac{2 (- x) + 2 (- 14)}{(x + 4) (x - 4)} = 0$

Step 2.10.3

Factor $2$ out of $2 (- x) + 2 (- 14)$ .

$\frac{2 (- x - 14)}{(x + 4) (x - 4)} = 0$

Step 2.11

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

$\frac{2 (- (x) - 14)}{(x + 4) (x - 4)} = 0$

Step 2.12

Rewrite $- 14$ as $- 1 (14)$ .

$\frac{2 (- (x) - 1 (14))}{(x + 4) (x - 4)} = 0$

Step 2.13

Factor $- 1$ out of $- (x) - 1 (14)$ .

$\frac{2 (- (x + 14))}{(x + 4) (x - 4)} = 0$

Step 2.14

Rewrite $- (x + 14)$ as $- 1 (x + 14)$ .

$\frac{2 (- 1 (x + 14))}{(x + 4) (x - 4)} = 0$

Step 2.15

Move the negative in front of the fraction.

$- \frac{2 (x + 14)}{(x + 4) (x - 4)} = 0$

Step 3

Set the denominator in $\frac{2 (x + 14)}{(x + 4) (x - 4)}$ equal to $0$ to find where the expression is undefined.

$(x + 4) (x - 4) = 0$

Step 4

Solve for $x$ .

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Step 4.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 4 = 0$

$x - 4 = 0$

Step 4.2

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 4.2.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 4.2.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 4.3

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 4.3.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 4.3.2

Add $4$ to both sides of the equation.

$x = 4$

Step 4.4

The final solution is all the values that make $(x + 4) (x - 4) = 0$ true.

$x = - 4, 4$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$x = - 4, x = 4$

Step 6