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

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

Evaluate

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

Short Solution Steps

Factor x^{2}+x-6.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -x-3.

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

Cancel out x+3 in both numerator and denominator.

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

Expand

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

Short Solution Steps

Factor x^{2}+x-6.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Extract the negative sign in -x-3.

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

Cancel out x+3 in both numerator and denominator.

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