Solve 3/x-frac{x-2/frac{3-x{2}}}{3x+2}

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

Evaluate

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

Short Solution Steps

Divide x-2 by \frac{3-x}{2} by multiplying x-2 by the reciprocal of \frac{3-x}{2}.

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

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

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

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

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

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

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

Combine like terms in -3x+9-2x^{2}+4x.

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

Express \frac{\frac{x+9-2x^{2}}{x\left(-x+3\right)}}{3x+2} as a single fraction.

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

Use the distributive property to multiply x by -x+3.

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

Apply the distributive property by multiplying each term of -x^{2}+3x by each term of 3x+2.

$$\frac{x+9-2x^{2}}{-3x^{3}-2x^{2}+9x^{2}+6x}$$

Combine -2x^{2} and 9x^{2} to get 7x^{2}.

$$\frac{x+9-2x^{2}}{-3x^{3}+7x^{2}+6x}$$

Expand

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

Short Solution Steps

Divide x-2 by \frac{3-x}{2} by multiplying x-2 by the reciprocal of \frac{3-x}{2}.

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

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

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

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

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

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

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

Combine like terms in -3x+9-2x^{2}+4x.

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

Express \frac{\frac{x+9-2x^{2}}{x\left(-x+3\right)}}{3x+2} as a single fraction.

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

Use the distributive property to multiply x by -x+3.

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

Apply the distributive property by multiplying each term of -x^{2}+3x by each term of 3x+2.

$$\frac{x+9-2x^{2}}{-3x^{3}-2x^{2}+9x^{2}+6x}$$

Combine -2x^{2} and 9x^{2} to get 7x^{2}.

$$\frac{x+9-2x^{2}}{-3x^{3}+7x^{2}+6x}$$