Solve 3x-2/2x^2-5x-3-x-2/6x^2+x-1-2x+2/3x^2-10x+3

$$\frac { 3 x - 2 } { 2 x ^ { 2 } - 5 x - 3 } - \frac { x - 2 } { 6 x ^ { 2 } + x - 1 } - \frac { 2 x + 2 } { 3 x ^ { 2 } - 10 x + 3 }$$

Evaluate

$$\frac{2}{3x-1}$$

Short Solution Steps

Factor 2x^{2}-5x-3. Factor 6x^{2}+x-1.

$$\frac{3x-2}{\left(x-3\right)\left(2x+1\right)}-\frac{x-2}{\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{\left(3x-2\right)\left(3x-1\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{\left(x-2\right)\left(x-3\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{\left(3x-2\right)\left(3x-1\right)-\left(x-2\right)\left(x-3\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{9x^{2}-3x-6x+2-x^{2}+3x+2x-6}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Combine like terms in 9x^{2}-3x-6x+2-x^{2}+3x+2x-6.

$$\frac{8x^{2}-4x-4}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Factor the expressions that are not already factored in \frac{8x^{2}-4x-4}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}.

$$\frac{4\left(x-1\right)\left(2x+1\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Cancel out 2x+1 in both numerator and denominator.

$$\frac{4\left(x-1\right)}{\left(x-3\right)\left(3x-1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Factor 3x^{2}-10x+3.

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

Since \frac{4\left(x-1\right)}{\left(x-3\right)\left(3x-1\right)} and \frac{2x+2}{\left(x-3\right)\left(3x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(x-1\right)-\left(2x+2\right).

$$\frac{4x-4-2x-2}{\left(x-3\right)\left(3x-1\right)}$$

Combine like terms in 4x-4-2x-2.

$$\frac{2x-6}{\left(x-3\right)\left(3x-1\right)}$$

Factor the expressions that are not already factored in \frac{2x-6}{\left(x-3\right)\left(3x-1\right)}.

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

Cancel out x-3 in both numerator and denominator.

$$\frac{2}{3x-1}$$

Expand

$$\frac{2}{3x-1}$$

Short Solution Steps

Factor 2x^{2}-5x-3. Factor 6x^{2}+x-1.

$$\frac{3x-2}{\left(x-3\right)\left(2x+1\right)}-\frac{x-2}{\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{\left(3x-2\right)\left(3x-1\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{\left(x-2\right)\left(x-3\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{\left(3x-2\right)\left(3x-1\right)-\left(x-2\right)\left(x-3\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

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

$$\frac{9x^{2}-3x-6x+2-x^{2}+3x+2x-6}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Combine like terms in 9x^{2}-3x-6x+2-x^{2}+3x+2x-6.

$$\frac{8x^{2}-4x-4}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Factor the expressions that are not already factored in \frac{8x^{2}-4x-4}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}.

$$\frac{4\left(x-1\right)\left(2x+1\right)}{\left(x-3\right)\left(3x-1\right)\left(2x+1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Cancel out 2x+1 in both numerator and denominator.

$$\frac{4\left(x-1\right)}{\left(x-3\right)\left(3x-1\right)}-\frac{2x+2}{3x^{2}-10x+3}$$

Factor 3x^{2}-10x+3.

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

Since \frac{4\left(x-1\right)}{\left(x-3\right)\left(3x-1\right)} and \frac{2x+2}{\left(x-3\right)\left(3x-1\right)} have the same denominator, subtract them by subtracting their numerators.

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

Do the multiplications in 4\left(x-1\right)-\left(2x+2\right).

$$\frac{4x-4-2x-2}{\left(x-3\right)\left(3x-1\right)}$$

Combine like terms in 4x-4-2x-2.

$$\frac{2x-6}{\left(x-3\right)\left(3x-1\right)}$$

Factor the expressions that are not already factored in \frac{2x-6}{\left(x-3\right)\left(3x-1\right)}.

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

Cancel out x-3 in both numerator and denominator.

$$\frac{2}{3x-1}$$