Solve 5r/3r-2+2/r-1

$$\frac { 5 r } { 3 r - 2 } + \frac { 2 } { r - 1 }$$

Evaluate

$$\frac{\left(5r-4\right)\left(r+1\right)}{\left(r-1\right)\left(3r-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3r-2 and r-1 is \left(r-1\right)\left(3r-2\right). Multiply \frac{5r}{3r-2} times \frac{r-1}{r-1}. Multiply \frac{2}{r-1} times \frac{3r-2}{3r-2}.

$$\frac{5r\left(r-1\right)}{\left(r-1\right)\left(3r-2\right)}+\frac{2\left(3r-2\right)}{\left(r-1\right)\left(3r-2\right)}$$

Since \frac{5r\left(r-1\right)}{\left(r-1\right)\left(3r-2\right)} and \frac{2\left(3r-2\right)}{\left(r-1\right)\left(3r-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{5r\left(r-1\right)+2\left(3r-2\right)}{\left(r-1\right)\left(3r-2\right)}$$

Do the multiplications in 5r\left(r-1\right)+2\left(3r-2\right).

$$\frac{5r^{2}-5r+6r-4}{\left(r-1\right)\left(3r-2\right)}$$

Combine like terms in 5r^{2}-5r+6r-4.

$$\frac{5r^{2}+r-4}{\left(r-1\right)\left(3r-2\right)}$$

Expand \left(r-1\right)\left(3r-2\right).

$$\frac{5r^{2}+r-4}{3r^{2}-5r+2}$$

Differentiate w.r.t. r

$$\frac{2\left(-14r^{2}+22r-9\right)}{\left(\left(r-1\right)\left(3r-2\right)\right)^{2}}$$