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

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

Evaluate

$$\frac{11x^{2}-23x+6}{\left(x-1\right)\left(3x-5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and 3x-5 is \left(x-1\right)\left(3x-5\right). Multiply \frac{4x-1}{x-1} times \frac{3x-5}{3x-5}. Multiply \frac{x+1}{3x-5} times \frac{x-1}{x-1}.

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

Since \frac{\left(4x-1\right)\left(3x-5\right)}{\left(x-1\right)\left(3x-5\right)} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(3x-5\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(4x-1\right)\left(3x-5\right)-\left(x+1\right)\left(x-1\right).

$$\frac{12x^{2}-20x-3x+5-x^{2}+x-x+1}{\left(x-1\right)\left(3x-5\right)}$$

Combine like terms in 12x^{2}-20x-3x+5-x^{2}+x-x+1.

$$\frac{11x^{2}-23x+6}{\left(x-1\right)\left(3x-5\right)}$$

Expand \left(x-1\right)\left(3x-5\right).

$$\frac{11x^{2}-23x+6}{3x^{2}-8x+5}$$

Expand

$$\frac{11x^{2}-23x+6}{\left(x-1\right)\left(3x-5\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x-1 and 3x-5 is \left(x-1\right)\left(3x-5\right). Multiply \frac{4x-1}{x-1} times \frac{3x-5}{3x-5}. Multiply \frac{x+1}{3x-5} times \frac{x-1}{x-1}.

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

Since \frac{\left(4x-1\right)\left(3x-5\right)}{\left(x-1\right)\left(3x-5\right)} and \frac{\left(x+1\right)\left(x-1\right)}{\left(x-1\right)\left(3x-5\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in \left(4x-1\right)\left(3x-5\right)-\left(x+1\right)\left(x-1\right).

$$\frac{12x^{2}-20x-3x+5-x^{2}+x-x+1}{\left(x-1\right)\left(3x-5\right)}$$

Combine like terms in 12x^{2}-20x-3x+5-x^{2}+x-x+1.

$$\frac{11x^{2}-23x+6}{\left(x-1\right)\left(3x-5\right)}$$

Expand \left(x-1\right)\left(3x-5\right).

$$\frac{11x^{2}-23x+6}{3x^{2}-8x+5}$$