Solve 4/(x-9)(x+6)-5/(x+6)(x+4)

$$\frac { 4 } { ( x - 9 ) ( x + 6 ) } - \frac { 5 } { ( x + 6 ) ( x + 4 ) }$$

Evaluate

$$\frac{61-x}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-9\right)\left(x+6\right) and \left(x+6\right)\left(x+4\right) is \left(x-9\right)\left(x+4\right)\left(x+6\right). Multiply \frac{4}{\left(x-9\right)\left(x+6\right)} times \frac{x+4}{x+4}. Multiply \frac{5}{\left(x+6\right)\left(x+4\right)} times \frac{x-9}{x-9}.

$$\frac{4\left(x+4\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}-\frac{5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Since \frac{4\left(x+4\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)} and \frac{5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(x+4\right)-5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Do the multiplications in 4\left(x+4\right)-5\left(x-9\right).

$$\frac{4x+16-5x+45}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Combine like terms in 4x+16-5x+45.

$$\frac{-x+61}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Expand \left(x-9\right)\left(x+4\right)\left(x+6\right).

$$\frac{-x+61}{x^{3}+x^{2}-66x-216}$$

Expand

$$\frac{61-x}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(x-9\right)\left(x+6\right) and \left(x+6\right)\left(x+4\right) is \left(x-9\right)\left(x+4\right)\left(x+6\right). Multiply \frac{4}{\left(x-9\right)\left(x+6\right)} times \frac{x+4}{x+4}. Multiply \frac{5}{\left(x+6\right)\left(x+4\right)} times \frac{x-9}{x-9}.

$$\frac{4\left(x+4\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}-\frac{5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Since \frac{4\left(x+4\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)} and \frac{5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4\left(x+4\right)-5\left(x-9\right)}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Do the multiplications in 4\left(x+4\right)-5\left(x-9\right).

$$\frac{4x+16-5x+45}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Combine like terms in 4x+16-5x+45.

$$\frac{-x+61}{\left(x-9\right)\left(x+4\right)\left(x+6\right)}$$

Expand \left(x-9\right)\left(x+4\right)\left(x+6\right).

$$\frac{-x+61}{x^{3}+x^{2}-66x-216}$$