Solve x/5x-6-x+3/6x

$$\frac { x } { 5 x - 6 } - \frac { x + 3 } { 6 x }$$

Evaluate

$$\frac{\left(x-6\right)\left(x-3\right)}{6x\left(5x-6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5x-6 and 6x is 6x\left(5x-6\right). Multiply \frac{x}{5x-6} times \frac{6x}{6x}. Multiply \frac{x+3}{6x} times \frac{5x-6}{5x-6}.

$$\frac{x\times 6x}{6x\left(5x-6\right)}-\frac{\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)}$$

Since \frac{x\times 6x}{6x\left(5x-6\right)} and \frac{\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 6x-\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)}$$

Do the multiplications in x\times 6x-\left(x+3\right)\left(5x-6\right).

$$\frac{6x^{2}-5x^{2}+6x-15x+18}{6x\left(5x-6\right)}$$

Combine like terms in 6x^{2}-5x^{2}+6x-15x+18.

$$\frac{x^{2}-9x+18}{6x\left(5x-6\right)}$$

Expand 6x\left(5x-6\right).

$$\frac{x^{2}-9x+18}{30x^{2}-36x}$$

Expand

$$\frac{x^{2}-9x+18}{6x\left(5x-6\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 5x-6 and 6x is 6x\left(5x-6\right). Multiply \frac{x}{5x-6} times \frac{6x}{6x}. Multiply \frac{x+3}{6x} times \frac{5x-6}{5x-6}.

$$\frac{x\times 6x}{6x\left(5x-6\right)}-\frac{\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)}$$

Since \frac{x\times 6x}{6x\left(5x-6\right)} and \frac{\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x\times 6x-\left(x+3\right)\left(5x-6\right)}{6x\left(5x-6\right)}$$

Do the multiplications in x\times 6x-\left(x+3\right)\left(5x-6\right).

$$\frac{6x^{2}-5x^{2}+6x-15x+18}{6x\left(5x-6\right)}$$

Combine like terms in 6x^{2}-5x^{2}+6x-15x+18.

$$\frac{x^{2}-9x+18}{6x\left(5x-6\right)}$$

Expand 6x\left(5x-6\right).

$$\frac{x^{2}-9x+18}{30x^{2}-36x}$$