Solve 3/3x-2-1/x-4+5

$$\frac { 3 } { 3 x - 2 } - \frac { 1 } { x - 4 } + 5$$

Evaluate

$$\frac{5\left(3x^{2}-14x+6\right)}{\left(x-4\right)\left(3x-2\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3x-2 and x-4 is \left(x-4\right)\left(3x-2\right). Multiply \frac{3}{3x-2} times \frac{x-4}{x-4}. Multiply \frac{1}{x-4} times \frac{3x-2}{3x-2}.

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

Since \frac{3\left(x-4\right)}{\left(x-4\right)\left(3x-2\right)} and \frac{3x-2}{\left(x-4\right)\left(3x-2\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 3\left(x-4\right)-\left(3x-2\right).

$$\frac{3x-12-3x+2}{\left(x-4\right)\left(3x-2\right)}+5$$

Combine like terms in 3x-12-3x+2.

$$\frac{-10}{\left(x-4\right)\left(3x-2\right)}+5$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 5 times \frac{\left(x-4\right)\left(3x-2\right)}{\left(x-4\right)\left(3x-2\right)}.

$$\frac{-10}{\left(x-4\right)\left(3x-2\right)}+\frac{5\left(x-4\right)\left(3x-2\right)}{\left(x-4\right)\left(3x-2\right)}$$

Since \frac{-10}{\left(x-4\right)\left(3x-2\right)} and \frac{5\left(x-4\right)\left(3x-2\right)}{\left(x-4\right)\left(3x-2\right)} have the same denominator, add them by adding their numerators.

$$\frac{-10+5\left(x-4\right)\left(3x-2\right)}{\left(x-4\right)\left(3x-2\right)}$$

Do the multiplications in -10+5\left(x-4\right)\left(3x-2\right).

$$\frac{-10+15x^{2}-10x-60x+40}{\left(x-4\right)\left(3x-2\right)}$$

Combine like terms in -10+15x^{2}-10x-60x+40.

$$\frac{30+15x^{2}-70x}{\left(x-4\right)\left(3x-2\right)}$$

Expand \left(x-4\right)\left(3x-2\right).

$$\frac{30+15x^{2}-70x}{3x^{2}-14x+8}$$

Differentiate w.r.t. x

$$\frac{20\left(3x-7\right)}{\left(\left(x-4\right)\left(3x-2\right)\right)^{2}}$$