Solve x+2/3x-1+x+1/3-2x+4x^2+6x+3/6x^2-11x+3

$$\frac { x + 2 } { 3 x - 1 } + \frac { x + 1 } { 3 - 2 x } + \frac { 4 x ^ { 2 } + 6 x + 3 } { 6 x ^ { 2 } - 11 x + 3 }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\left(x+2\right)\left(-2x+3\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{-2x^{2}+3x-4x+6+3x^{2}-x+3x-1}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Combine like terms in -2x^{2}+3x-4x+6+3x^{2}-x+3x-1.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Factor 6x^{2}-11x+3.

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

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

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

Since \frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)} and \frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(x^{2}+x+5\right)+4x^{2}+6x+3.

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

Combine like terms in -x^{2}-x-5+4x^{2}+6x+3.

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

Factor the expressions that are not already factored in \frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}.

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

Cancel out 3x-1 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

$$\frac{\left(x+2\right)\left(-2x+3\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{\left(x+2\right)\left(-2x+3\right)+\left(x+1\right)\left(3x-1\right)}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

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

$$\frac{-2x^{2}+3x-4x+6+3x^{2}-x+3x-1}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Combine like terms in -2x^{2}+3x-4x+6+3x^{2}-x+3x-1.

$$\frac{x^{2}+x+5}{\left(3x-1\right)\left(-2x+3\right)}+\frac{4x^{2}+6x+3}{6x^{2}-11x+3}$$

Factor 6x^{2}-11x+3.

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

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

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

Since \frac{-\left(x^{2}+x+5\right)}{\left(2x-3\right)\left(3x-1\right)} and \frac{4x^{2}+6x+3}{\left(2x-3\right)\left(3x-1\right)} have the same denominator, add them by adding their numerators.

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

Do the multiplications in -\left(x^{2}+x+5\right)+4x^{2}+6x+3.

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

Combine like terms in -x^{2}-x-5+4x^{2}+6x+3.

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

Factor the expressions that are not already factored in \frac{3x^{2}+5x-2}{\left(2x-3\right)\left(3x-1\right)}.

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

Cancel out 3x-1 in both numerator and denominator.

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