Solve 18/7x-14-2/x-1+3/8x+40

$$\frac { 18 } { 7 x - 14 } - \frac { 2 } { x - 1 } + \frac { 3 } { 8 x + 40 }$$

Evaluate

$$\frac{53x^{2}+177x+442}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Short Solution Steps

Factor 7x-14.

$$\frac{18}{7\left(x-2\right)}-\frac{2}{x-1}+\frac{3}{8x+40}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7\left(x-2\right) and x-1 is 7\left(x-2\right)\left(x-1\right). Multiply \frac{18}{7\left(x-2\right)} times \frac{x-1}{x-1}. Multiply \frac{2}{x-1} times \frac{7\left(x-2\right)}{7\left(x-2\right)}.

$$\frac{18\left(x-1\right)}{7\left(x-2\right)\left(x-1\right)}-\frac{2\times 7\left(x-2\right)}{7\left(x-2\right)\left(x-1\right)}+\frac{3}{8x+40}$$

Since \frac{18\left(x-1\right)}{7\left(x-2\right)\left(x-1\right)} and \frac{2\times 7\left(x-2\right)}{7\left(x-2\right)\left(x-1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{18\left(x-1\right)-2\times 7\left(x-2\right)}{7\left(x-2\right)\left(x-1\right)}+\frac{3}{8x+40}$$

Do the multiplications in 18\left(x-1\right)-2\times 7\left(x-2\right).

$$\frac{18x-18-14x+28}{7\left(x-2\right)\left(x-1\right)}+\frac{3}{8x+40}$$

Combine like terms in 18x-18-14x+28.

$$\frac{4x+10}{7\left(x-2\right)\left(x-1\right)}+\frac{3}{8x+40}$$

Factor 8x+40.

$$\frac{4x+10}{7\left(x-2\right)\left(x-1\right)}+\frac{3}{8\left(x+5\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 7\left(x-2\right)\left(x-1\right) and 8\left(x+5\right) is 56\left(x-2\right)\left(x-1\right)\left(x+5\right). Multiply \frac{4x+10}{7\left(x-2\right)\left(x-1\right)} times \frac{8\left(x+5\right)}{8\left(x+5\right)}. Multiply \frac{3}{8\left(x+5\right)} times \frac{7\left(x-2\right)\left(x-1\right)}{7\left(x-2\right)\left(x-1\right)}.

$$\frac{\left(4x+10\right)\times 8\left(x+5\right)}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}+\frac{3\times 7\left(x-2\right)\left(x-1\right)}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Since \frac{\left(4x+10\right)\times 8\left(x+5\right)}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)} and \frac{3\times 7\left(x-2\right)\left(x-1\right)}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(4x+10\right)\times 8\left(x+5\right)+3\times 7\left(x-2\right)\left(x-1\right)}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Do the multiplications in \left(4x+10\right)\times 8\left(x+5\right)+3\times 7\left(x-2\right)\left(x-1\right).

$$\frac{32x^{2}+160x+80x+400+21x^{2}-21x-42x+42}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Combine like terms in 32x^{2}+160x+80x+400+21x^{2}-21x-42x+42.

$$\frac{53x^{2}+177x+442}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$

Expand 56\left(x-2\right)\left(x-1\right)\left(x+5\right).

$$\frac{53x^{2}+177x+442}{56x^{3}+112x^{2}-728x+560}$$

Factor

$$\frac{53x^{2}+177x+442}{56\left(x-2\right)\left(x-1\right)\left(x+5\right)}$$