Solve 9/(x-6)+10/(6-x)(x+4)+1/(x+3)(x+4)

$$\frac { 9 } { ( x - 6 ) } + \frac { 10 } { ( 6 - x ) ( x + 4 ) } + \frac { 1 } { ( x + 3 ) ( x + 4 ) }$$

Evaluate

$$-\frac{9\left(x+2\right)}{\left(6-x\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-6 and \left(6-x\right)\left(x+4\right) is \left(x-6\right)\left(x+4\right). Multiply \frac{9}{x-6} times \frac{x+4}{x+4}. Multiply \frac{10}{\left(6-x\right)\left(x+4\right)} times \frac{-1}{-1}.

$$\frac{9\left(x+4\right)}{\left(x-6\right)\left(x+4\right)}+\frac{10\left(-1\right)}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Since \frac{9\left(x+4\right)}{\left(x-6\right)\left(x+4\right)} and \frac{10\left(-1\right)}{\left(x-6\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{9\left(x+4\right)+10\left(-1\right)}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in 9\left(x+4\right)+10\left(-1\right).

$$\frac{9x+36-10}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x+36-10.

$$\frac{9x+26}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+4\right) and \left(x+3\right)\left(x+4\right) is \left(x-6\right)\left(x+3\right)\left(x+4\right). Multiply \frac{9x+26}{\left(x-6\right)\left(x+4\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{x-6}{x-6}.

$$\frac{\left(9x+26\right)\left(x+3\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}+\frac{x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(9x+26\right)\left(x+3\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)} and \frac{x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x+26\right)\left(x+3\right)+x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(9x+26\right)\left(x+3\right)+x-6.

$$\frac{9x^{2}+27x+26x+78+x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x^{2}+27x+26x+78+x-6.

$$\frac{9x^{2}+54x+72}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{9x^{2}+54x+72}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{9\left(x+2\right)\left(x+4\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{9\left(x+2\right)}{\left(x-6\right)\left(x+3\right)}$$

Expand \left(x-6\right)\left(x+3\right).

$$\frac{9\left(x+2\right)}{x^{2}-3x-18}$$

Use the distributive property to multiply 9 by x+2.

$$\frac{9x+18}{x^{2}-3x-18}$$

Expand

$$\frac{9\left(x+2\right)}{\left(x-6\right)\left(x+3\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-6 and \left(6-x\right)\left(x+4\right) is \left(x-6\right)\left(x+4\right). Multiply \frac{9}{x-6} times \frac{x+4}{x+4}. Multiply \frac{10}{\left(6-x\right)\left(x+4\right)} times \frac{-1}{-1}.

$$\frac{9\left(x+4\right)}{\left(x-6\right)\left(x+4\right)}+\frac{10\left(-1\right)}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Since \frac{9\left(x+4\right)}{\left(x-6\right)\left(x+4\right)} and \frac{10\left(-1\right)}{\left(x-6\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{9\left(x+4\right)+10\left(-1\right)}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in 9\left(x+4\right)+10\left(-1\right).

$$\frac{9x+36-10}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x+36-10.

$$\frac{9x+26}{\left(x-6\right)\left(x+4\right)}+\frac{1}{\left(x+3\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-6\right)\left(x+4\right) and \left(x+3\right)\left(x+4\right) is \left(x-6\right)\left(x+3\right)\left(x+4\right). Multiply \frac{9x+26}{\left(x-6\right)\left(x+4\right)} times \frac{x+3}{x+3}. Multiply \frac{1}{\left(x+3\right)\left(x+4\right)} times \frac{x-6}{x-6}.

$$\frac{\left(9x+26\right)\left(x+3\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}+\frac{x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Since \frac{\left(9x+26\right)\left(x+3\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)} and \frac{x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(9x+26\right)\left(x+3\right)+x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Do the multiplications in \left(9x+26\right)\left(x+3\right)+x-6.

$$\frac{9x^{2}+27x+26x+78+x-6}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Combine like terms in 9x^{2}+27x+26x+78+x-6.

$$\frac{9x^{2}+54x+72}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Factor the expressions that are not already factored in \frac{9x^{2}+54x+72}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}.

$$\frac{9\left(x+2\right)\left(x+4\right)}{\left(x-6\right)\left(x+3\right)\left(x+4\right)}$$

Cancel out x+4 in both numerator and denominator.

$$\frac{9\left(x+2\right)}{\left(x-6\right)\left(x+3\right)}$$

Expand \left(x-6\right)\left(x+3\right).

$$\frac{9\left(x+2\right)}{x^{2}-3x-18}$$

Use the distributive property to multiply 9 by x+2.

$$\frac{9x+18}{x^{2}-3x-18}$$