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

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

Evaluate

$$\frac{2r+7}{r\left(r+2\right)\left(r+3\right)}$$

Short Solution Steps

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

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

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

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

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

$$\frac{r^{2}+2r+2r+4-r^{2}-3r+r+3}{r\left(r+2\right)\left(r+3\right)}$$

Combine like terms in r^{2}+2r+2r+4-r^{2}-3r+r+3.

$$\frac{2r+7}{r\left(r+2\right)\left(r+3\right)}$$

Expand r\left(r+2\right)\left(r+3\right).

$$\frac{2r+7}{r^{3}+5r^{2}+6r}$$

Expand

$$\frac{2r+7}{r\left(r+2\right)\left(r+3\right)}$$

Short Solution Steps

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

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

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

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

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

$$\frac{r^{2}+2r+2r+4-r^{2}-3r+r+3}{r\left(r+2\right)\left(r+3\right)}$$

Combine like terms in r^{2}+2r+2r+4-r^{2}-3r+r+3.

$$\frac{2r+7}{r\left(r+2\right)\left(r+3\right)}$$

Expand r\left(r+2\right)\left(r+3\right).

$$\frac{2r+7}{r^{3}+5r^{2}+6r}$$