Solve x+1/2x-8-5x+2/3x+12*1+18/2x^2-32

$$\frac { x + 1 } { 2 x - 8 } - \frac { 5 x + 2 } { 3 x + 12 } \cdot 1 + \frac { 18 } { 2 x ^ { 2 } - 32 }$$

Evaluate

$$\frac{82+51x-7x^{2}}{6\left(x^{2}-16\right)}$$

Short Solution Steps

Express \frac{5x+2}{3x+12}\times 1 as a single fraction.

$$\frac{x+1}{2x-8}-\frac{5x+2}{3x+12}+\frac{18}{2x^{2}-32}$$

Factor 2x-8. Factor 3x+12.

$$\frac{x+1}{2\left(x-4\right)}-\frac{5x+2}{3\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-4\right) and 3\left(x+4\right) is 6\left(x-4\right)\left(x+4\right). Multiply \frac{x+1}{2\left(x-4\right)} times \frac{3\left(x+4\right)}{3\left(x+4\right)}. Multiply \frac{5x+2}{3\left(x+4\right)} times \frac{2\left(x-4\right)}{2\left(x-4\right)}.

$$\frac{\left(x+1\right)\times 3\left(x+4\right)}{6\left(x-4\right)\left(x+4\right)}-\frac{\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Since \frac{\left(x+1\right)\times 3\left(x+4\right)}{6\left(x-4\right)\left(x+4\right)} and \frac{\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\times 3\left(x+4\right)-\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Do the multiplications in \left(x+1\right)\times 3\left(x+4\right)-\left(5x+2\right)\times 2\left(x-4\right).

$$\frac{3x^{2}+12x+3x+12-10x^{2}+40x-4x+16}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Combine like terms in 3x^{2}+12x+3x+12-10x^{2}+40x-4x+16.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Factor 2x^{2}-32.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6\left(x-4\right)\left(x+4\right) and 2\left(x-4\right)\left(x+4\right) is 6\left(x-4\right)\left(x+4\right). Multiply \frac{18}{2\left(x-4\right)\left(x+4\right)} times \frac{3}{3}.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18\times 3}{6\left(x-4\right)\left(x+4\right)}$$

Since \frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)} and \frac{18\times 3}{6\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+51x+28+18\times 3}{6\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in -7x^{2}+51x+28+18\times 3.

$$\frac{-7x^{2}+51x+28+54}{6\left(x-4\right)\left(x+4\right)}$$

Combine like terms in -7x^{2}+51x+28+54.

$$\frac{-7x^{2}+51x+82}{6\left(x-4\right)\left(x+4\right)}$$

Expand 6\left(x-4\right)\left(x+4\right).

$$\frac{-7x^{2}+51x+82}{6x^{2}-96}$$

Expand

$$-\frac{7x^{2}-51x-82}{6\left(x^{2}-16\right)}$$

Short Solution Steps

Express \frac{5x+2}{3x+12}\times 1 as a single fraction.

$$\frac{x+1}{2x-8}-\frac{5x+2}{3x+12}+\frac{18}{2x^{2}-32}$$

Factor 2x-8. Factor 3x+12.

$$\frac{x+1}{2\left(x-4\right)}-\frac{5x+2}{3\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(x-4\right) and 3\left(x+4\right) is 6\left(x-4\right)\left(x+4\right). Multiply \frac{x+1}{2\left(x-4\right)} times \frac{3\left(x+4\right)}{3\left(x+4\right)}. Multiply \frac{5x+2}{3\left(x+4\right)} times \frac{2\left(x-4\right)}{2\left(x-4\right)}.

$$\frac{\left(x+1\right)\times 3\left(x+4\right)}{6\left(x-4\right)\left(x+4\right)}-\frac{\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Since \frac{\left(x+1\right)\times 3\left(x+4\right)}{6\left(x-4\right)\left(x+4\right)} and \frac{\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(x+1\right)\times 3\left(x+4\right)-\left(5x+2\right)\times 2\left(x-4\right)}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Do the multiplications in \left(x+1\right)\times 3\left(x+4\right)-\left(5x+2\right)\times 2\left(x-4\right).

$$\frac{3x^{2}+12x+3x+12-10x^{2}+40x-4x+16}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Combine like terms in 3x^{2}+12x+3x+12-10x^{2}+40x-4x+16.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2x^{2}-32}$$

Factor 2x^{2}-32.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18}{2\left(x-4\right)\left(x+4\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6\left(x-4\right)\left(x+4\right) and 2\left(x-4\right)\left(x+4\right) is 6\left(x-4\right)\left(x+4\right). Multiply \frac{18}{2\left(x-4\right)\left(x+4\right)} times \frac{3}{3}.

$$\frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)}+\frac{18\times 3}{6\left(x-4\right)\left(x+4\right)}$$

Since \frac{-7x^{2}+51x+28}{6\left(x-4\right)\left(x+4\right)} and \frac{18\times 3}{6\left(x-4\right)\left(x+4\right)} have the same denominator, add them by adding their numerators.

$$\frac{-7x^{2}+51x+28+18\times 3}{6\left(x-4\right)\left(x+4\right)}$$

Do the multiplications in -7x^{2}+51x+28+18\times 3.

$$\frac{-7x^{2}+51x+28+54}{6\left(x-4\right)\left(x+4\right)}$$

Combine like terms in -7x^{2}+51x+28+54.

$$\frac{-7x^{2}+51x+82}{6\left(x-4\right)\left(x+4\right)}$$

Expand 6\left(x-4\right)\left(x+4\right).

$$\frac{-7x^{2}+51x+82}{6x^{2}-96}$$