Solve x+1/1-2x-x+3/4x-3+10x^2+7x-9/8x^2-10x+3

$$\frac { x + 1 } { 1 - 2 x } - \frac { x + 3 } { 4 x - 3 } + \frac { 10 x ^ { 2 } + 7 x - 9 } { 8 x ^ { 2 } - 10 x + 3 }$$

Evaluate

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

Short Solution Steps

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

$$\frac{\left(x+1\right)\left(4x-3\right)}{\left(4x-3\right)\left(-2x+1\right)}-\frac{\left(x+3\right)\left(-2x+1\right)}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

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

$$\frac{\left(x+1\right)\left(4x-3\right)-\left(x+3\right)\left(-2x+1\right)}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

Do the multiplications in \left(x+1\right)\left(4x-3\right)-\left(x+3\right)\left(-2x+1\right).

$$\frac{4x^{2}-3x+4x-3+2x^{2}-x+6x-3}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

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

$$\frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

Factor 8x^{2}-10x+3.

$$\frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4x-3\right)\left(-2x+1\right) and \left(2x-1\right)\left(4x-3\right) is \left(2x-1\right)\left(4x-3\right). Multiply \frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)} times \frac{-1}{-1}.

$$\frac{-\left(6x^{2}+6x-6\right)}{\left(2x-1\right)\left(4x-3\right)}+\frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Since \frac{-\left(6x^{2}+6x-6\right)}{\left(2x-1\right)\left(4x-3\right)} and \frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(6x^{2}+6x-6\right)+10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Do the multiplications in -\left(6x^{2}+6x-6\right)+10x^{2}+7x-9.

$$\frac{-6x^{2}-6x+6+10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Combine like terms in -6x^{2}-6x+6+10x^{2}+7x-9.

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

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

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

Cancel out 4x-3 in both numerator and denominator.

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

Expand

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

Short Solution Steps

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

$$\frac{\left(x+1\right)\left(4x-3\right)}{\left(4x-3\right)\left(-2x+1\right)}-\frac{\left(x+3\right)\left(-2x+1\right)}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

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

$$\frac{\left(x+1\right)\left(4x-3\right)-\left(x+3\right)\left(-2x+1\right)}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

Do the multiplications in \left(x+1\right)\left(4x-3\right)-\left(x+3\right)\left(-2x+1\right).

$$\frac{4x^{2}-3x+4x-3+2x^{2}-x+6x-3}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

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

$$\frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{8x^{2}-10x+3}$$

Factor 8x^{2}-10x+3.

$$\frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)}+\frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \left(4x-3\right)\left(-2x+1\right) and \left(2x-1\right)\left(4x-3\right) is \left(2x-1\right)\left(4x-3\right). Multiply \frac{6x^{2}+6x-6}{\left(4x-3\right)\left(-2x+1\right)} times \frac{-1}{-1}.

$$\frac{-\left(6x^{2}+6x-6\right)}{\left(2x-1\right)\left(4x-3\right)}+\frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Since \frac{-\left(6x^{2}+6x-6\right)}{\left(2x-1\right)\left(4x-3\right)} and \frac{10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)} have the same denominator, add them by adding their numerators.

$$\frac{-\left(6x^{2}+6x-6\right)+10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Do the multiplications in -\left(6x^{2}+6x-6\right)+10x^{2}+7x-9.

$$\frac{-6x^{2}-6x+6+10x^{2}+7x-9}{\left(2x-1\right)\left(4x-3\right)}$$

Combine like terms in -6x^{2}-6x+6+10x^{2}+7x-9.

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

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

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

Cancel out 4x-3 in both numerator and denominator.

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