Solve 5/x+8-2/x+6/x+9

$$\frac { \frac { 5 } { x + 8 } - \frac { 2 } { x + 6 } } { x + 9 }$$

Evaluate

$$\frac{3x+14}{\left(x+6\right)\left(x+8\right)\left(x+9\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+8 and x+6 is \left(x+6\right)\left(x+8\right). Multiply \frac{5}{x+8} times \frac{x+6}{x+6}. Multiply \frac{2}{x+6} times \frac{x+8}{x+8}.

$$\frac{\frac{5\left(x+6\right)}{\left(x+6\right)\left(x+8\right)}-\frac{2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Since \frac{5\left(x+6\right)}{\left(x+6\right)\left(x+8\right)} and \frac{2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5\left(x+6\right)-2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Do the multiplications in 5\left(x+6\right)-2\left(x+8\right).

$$\frac{\frac{5x+30-2x-16}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Combine like terms in 5x+30-2x-16.

$$\frac{\frac{3x+14}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Express \frac{\frac{3x+14}{\left(x+6\right)\left(x+8\right)}}{x+9} as a single fraction.

$$\frac{3x+14}{\left(x+6\right)\left(x+8\right)\left(x+9\right)}$$

Apply the distributive property by multiplying each term of x+6 by each term of x+8.

$$\frac{3x+14}{\left(x^{2}+8x+6x+48\right)\left(x+9\right)}$$

Combine 8x and 6x to get 14x.

$$\frac{3x+14}{\left(x^{2}+14x+48\right)\left(x+9\right)}$$

Apply the distributive property by multiplying each term of x^{2}+14x+48 by each term of x+9.

$$\frac{3x+14}{x^{3}+9x^{2}+14x^{2}+126x+48x+432}$$

Combine 9x^{2} and 14x^{2} to get 23x^{2}.

$$\frac{3x+14}{x^{3}+23x^{2}+126x+48x+432}$$

Combine 126x and 48x to get 174x.

$$\frac{3x+14}{x^{3}+23x^{2}+174x+432}$$

Expand

$$\frac{3x+14}{\left(x+6\right)\left(x+8\right)\left(x+9\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+8 and x+6 is \left(x+6\right)\left(x+8\right). Multiply \frac{5}{x+8} times \frac{x+6}{x+6}. Multiply \frac{2}{x+6} times \frac{x+8}{x+8}.

$$\frac{\frac{5\left(x+6\right)}{\left(x+6\right)\left(x+8\right)}-\frac{2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Since \frac{5\left(x+6\right)}{\left(x+6\right)\left(x+8\right)} and \frac{2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{5\left(x+6\right)-2\left(x+8\right)}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Do the multiplications in 5\left(x+6\right)-2\left(x+8\right).

$$\frac{\frac{5x+30-2x-16}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Combine like terms in 5x+30-2x-16.

$$\frac{\frac{3x+14}{\left(x+6\right)\left(x+8\right)}}{x+9}$$

Express \frac{\frac{3x+14}{\left(x+6\right)\left(x+8\right)}}{x+9} as a single fraction.

$$\frac{3x+14}{\left(x+6\right)\left(x+8\right)\left(x+9\right)}$$

Apply the distributive property by multiplying each term of x+6 by each term of x+8.

$$\frac{3x+14}{\left(x^{2}+8x+6x+48\right)\left(x+9\right)}$$

Combine 8x and 6x to get 14x.

$$\frac{3x+14}{\left(x^{2}+14x+48\right)\left(x+9\right)}$$

Apply the distributive property by multiplying each term of x^{2}+14x+48 by each term of x+9.

$$\frac{3x+14}{x^{3}+9x^{2}+14x^{2}+126x+48x+432}$$

Combine 9x^{2} and 14x^{2} to get 23x^{2}.

$$\frac{3x+14}{x^{3}+23x^{2}+126x+48x+432}$$

Combine 126x and 48x to get 174x.

$$\frac{3x+14}{x^{3}+23x^{2}+174x+432}$$