Solve E(x)=x+1/3x-6-(2/x-2+1/4-x^2):2x^2+3x/x^2+2x

$$E ( x ) = \frac { x + 1 } { 3 x - 6 } - ( \frac { 2 } { x - 2 } + \frac { 1 } { 4 - x ^ { 2 } } ) : \frac { 2 x ^ { 2 } + 3 x } { x ^ { 2 } + 2 x }$$

Evaluate

$$\frac{1}{3}\approx 0.333333333$$

Short Solution Steps

Factor 4-x^{2}.

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

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

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

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

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

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

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

Combine like terms in -2x-4+1.

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

Factor the expressions that are not already factored in \frac{2x^{2}+3x}{x^{2}+2x}.

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

Cancel out x in both numerator and denominator.

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

Divide \frac{-2x-3}{\left(x-2\right)\left(-x-2\right)} by \frac{2x+3}{x+2} by multiplying \frac{-2x-3}{\left(x-2\right)\left(-x-2\right)} by the reciprocal of \frac{2x+3}{x+2}.

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

Extract the negative sign in -2x-3. Extract the negative sign in x+2.

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

Cancel out \left(-x-2\right)\left(2x+3\right) in both numerator and denominator.

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

Multiply -1 and -1 to get 1.

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

Factor 3x-6.

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

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

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

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

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

Combine like terms in x+1-3.

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

Cancel out x-2 in both numerator and denominator.

$$\frac{1}{3}$$

Factor

$$\frac{1}{3} = 0.3333333333333333$$