Solve -3/r+6-r+2/2(r-1)

$$- \frac { 3 } { r + 6 } - \frac { r + 2 } { 2 ( r - 1 ) }$$

Evaluate

$$-\frac{r^{2}+14r+6}{2\left(r-1\right)\left(r+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r+6 and 2\left(r-1\right) is 2\left(r-1\right)\left(r+6\right). Multiply -\frac{3}{r+6} times \frac{2\left(r-1\right)}{2\left(r-1\right)}. Multiply \frac{r+2}{2\left(r-1\right)} times \frac{r+6}{r+6}.

$$-\frac{3\times 2\left(r-1\right)}{2\left(r-1\right)\left(r+6\right)}-\frac{\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)}$$

Since -\frac{3\times 2\left(r-1\right)}{2\left(r-1\right)\left(r+6\right)} and \frac{\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-3\times 2\left(r-1\right)-\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)}$$

Do the multiplications in -3\times 2\left(r-1\right)-\left(r+2\right)\left(r+6\right).

$$\frac{-6r+6-r^{2}-6r-2r-12}{2\left(r-1\right)\left(r+6\right)}$$

Combine like terms in -6r+6-r^{2}-6r-2r-12.

$$\frac{-14r-6-r^{2}}{2\left(r-1\right)\left(r+6\right)}$$

Expand 2\left(r-1\right)\left(r+6\right).

$$\frac{-14r-6-r^{2}}{2r^{2}+10r-12}$$

Expand

$$-\frac{r^{2}+14r+6}{2\left(r-1\right)\left(r+6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of r+6 and 2\left(r-1\right) is 2\left(r-1\right)\left(r+6\right). Multiply -\frac{3}{r+6} times \frac{2\left(r-1\right)}{2\left(r-1\right)}. Multiply \frac{r+2}{2\left(r-1\right)} times \frac{r+6}{r+6}.

$$-\frac{3\times 2\left(r-1\right)}{2\left(r-1\right)\left(r+6\right)}-\frac{\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)}$$

Since -\frac{3\times 2\left(r-1\right)}{2\left(r-1\right)\left(r+6\right)} and \frac{\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-3\times 2\left(r-1\right)-\left(r+2\right)\left(r+6\right)}{2\left(r-1\right)\left(r+6\right)}$$

Do the multiplications in -3\times 2\left(r-1\right)-\left(r+2\right)\left(r+6\right).

$$\frac{-6r+6-r^{2}-6r-2r-12}{2\left(r-1\right)\left(r+6\right)}$$

Combine like terms in -6r+6-r^{2}-6r-2r-12.

$$\frac{-14r-6-r^{2}}{2\left(r-1\right)\left(r+6\right)}$$

Expand 2\left(r-1\right)\left(r+6\right).

$$\frac{-14r-6-r^{2}}{2r^{2}+10r-12}$$