Solve 2x+3/x^2+x-6+x-7/x-2-3x-8/x+3

$$\frac { 2 x + 3 } { x ^ { 2 } + x - 6 } + \frac { x - 7 } { x - 2 } - \frac { 3 x - 8 } { x + 3 }$$

Evaluate

$$\frac{2\left(-x^{2}+6x-17\right)}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-6.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{x-7}{x-2}-\frac{3x-8}{x+3}$$

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-2 is \left(x-2\right)\left(x+3\right). Multiply \frac{x-7}{x-2} times \frac{x+3}{x+3}.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{\left(x-7\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

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

$$\frac{2x+3+\left(x-7\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Do the multiplications in 2x+3+\left(x-7\right)\left(x+3\right).

$$\frac{2x+3+x^{2}+3x-7x-21}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Combine like terms in 2x+3+x^{2}+3x-7x-21.

$$\frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

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{3x-8}{x+3} times \frac{x-2}{x-2}.

$$\frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Since \frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)} and \frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x-18+x^{2}-\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in -2x-18+x^{2}-\left(3x-8\right)\left(x-2\right).

$$\frac{-2x-18+x^{2}-3x^{2}+6x+8x-16}{\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -2x-18+x^{2}-3x^{2}+6x+8x-16.

$$\frac{12x-34-2x^{2}}{\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+3\right).

$$\frac{12x-34-2x^{2}}{x^{2}+x-6}$$

Expand

$$-\frac{2\left(x^{2}-6x+17\right)}{\left(x-2\right)\left(x+3\right)}$$

Short Solution Steps

Factor x^{2}+x-6.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{x-7}{x-2}-\frac{3x-8}{x+3}$$

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-2 is \left(x-2\right)\left(x+3\right). Multiply \frac{x-7}{x-2} times \frac{x+3}{x+3}.

$$\frac{2x+3}{\left(x-2\right)\left(x+3\right)}+\frac{\left(x-7\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

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

$$\frac{2x+3+\left(x-7\right)\left(x+3\right)}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Do the multiplications in 2x+3+\left(x-7\right)\left(x+3\right).

$$\frac{2x+3+x^{2}+3x-7x-21}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

Combine like terms in 2x+3+x^{2}+3x-7x-21.

$$\frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{3x-8}{x+3}$$

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{3x-8}{x+3} times \frac{x-2}{x-2}.

$$\frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)}-\frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Since \frac{-2x-18+x^{2}}{\left(x-2\right)\left(x+3\right)} and \frac{\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{-2x-18+x^{2}-\left(3x-8\right)\left(x-2\right)}{\left(x-2\right)\left(x+3\right)}$$

Do the multiplications in -2x-18+x^{2}-\left(3x-8\right)\left(x-2\right).

$$\frac{-2x-18+x^{2}-3x^{2}+6x+8x-16}{\left(x-2\right)\left(x+3\right)}$$

Combine like terms in -2x-18+x^{2}-3x^{2}+6x+8x-16.

$$\frac{12x-34-2x^{2}}{\left(x-2\right)\left(x+3\right)}$$

Expand \left(x-2\right)\left(x+3\right).

$$\frac{12x-34-2x^{2}}{x^{2}+x-6}$$