Basic Math Examples

$\frac{6}{y + 4} - \frac{8}{y - 4} = \frac{12}{y^{2}} - 16$

Step 1

Simplify $\frac{6}{y + 4} - \frac{8}{y - 4}$ .

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Step 1.1

To write $\frac{6}{y + 4}$ as a fraction with a common denominator, multiply by $\frac{y - 4}{y - 4}$ .

$\frac{6}{y + 4} \cdot \frac{y - 4}{y - 4} - \frac{8}{y - 4} = \frac{12}{y^{2}} - 16$

Step 1.2

To write $- \frac{8}{y - 4}$ as a fraction with a common denominator, multiply by $\frac{y + 4}{y + 4}$ .

$\frac{6}{y + 4} \cdot \frac{y - 4}{y - 4} - \frac{8}{y - 4} \cdot \frac{y + 4}{y + 4} = \frac{12}{y^{2}} - 16$

Step 1.3

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

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Step 1.3.1

Multiply $\frac{6}{y + 4}$ by $\frac{y - 4}{y - 4}$ .

$\frac{6 (y - 4)}{(y + 4) (y - 4)} - \frac{8}{y - 4} \cdot \frac{y + 4}{y + 4} = \frac{12}{y^{2}} - 16$

Step 1.3.2

Multiply $\frac{8}{y - 4}$ by $\frac{y + 4}{y + 4}$ .

$\frac{6 (y - 4)}{(y + 4) (y - 4)} - \frac{8 (y + 4)}{(y - 4) (y + 4)} = \frac{12}{y^{2}} - 16$

Step 1.3.3

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

$\frac{6 (y - 4)}{(y + 4) (y - 4)} - \frac{8 (y + 4)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.4

Combine the numerators over the common denominator.

$\frac{6 (y - 4) - 8 (y + 4)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5

Simplify the numerator.

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Step 1.5.1

Factor $2$ out of $6 (y - 4) - 8 (y + 4)$ .

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Step 1.5.1.1

Factor $2$ out of $6 (y - 4)$ .

$\frac{2 (3 (y - 4)) - 8 (y + 4)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.1.2

Factor $2$ out of $- 8 (y + 4)$ .

$\frac{2 (3 (y - 4)) + 2 (- 4 (y + 4))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.1.3

Factor $2$ out of $2 (3 (y - 4)) + 2 (- 4 (y + 4))$ .

$\frac{2 (3 (y - 4) - 4 (y + 4))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.2

Apply the distributive property.

$\frac{2 (3 y + 3 \cdot - 4 - 4 (y + 4))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.3

Multiply $3$ by $- 4$ .

$\frac{2 (3 y - 12 - 4 (y + 4))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.4

Apply the distributive property.

$\frac{2 (3 y - 12 - 4 y - 4 \cdot 4)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.5

Multiply $- 4$ by $4$ .

$\frac{2 (3 y - 12 - 4 y - 16)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.6

Subtract $4 y$ from $3 y$ .

$\frac{2 (- y - 12 - 16)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.5.7

Subtract $16$ from $- 12$ .

$\frac{2 (- y - 28)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.6

Simplify with factoring out.

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Step 1.6.1

Factor $- 1$ out of $- y$ .

$\frac{2 (- (y) - 28)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.6.2

Rewrite $- 28$ as $- 1 (28)$ .

$\frac{2 (- (y) - 1 (28))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.6.3

Factor $- 1$ out of $- (y) - 1 (28)$ .

$\frac{2 (- (y + 28))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.6.4

Simplify the expression.

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Step 1.6.4.1

Rewrite $- (y + 28)$ as $- 1 (y + 28)$ .

$\frac{2 (- 1 (y + 28))}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 1.6.4.2

Move the negative in front of the fraction.

$- \frac{2 (y + 28)}{(y + 4) (y - 4)} = \frac{12}{y^{2}} - 16$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$y \approx - 4.36748978, - 0.78370442, 0.77969017, 4.49650404$

Step 3