Solve 3/x-2+5/x-3-8/x+8=0

$$\frac { 3 } { x - 2 } + \frac { 5 } { x - 3 } - \frac { 8 } { x + 8 } = 0$$

Solve for x

$$x = \frac{40}{17} = 2\frac{6}{17} \approx 2.352941176$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -8,2,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-2\right)\left(x+8\right), the least common multiple of x-2,x-3,x+8.

$$\left(x-3\right)\left(x+8\right)\times 3+\left(x-2\right)\left(x+8\right)\times 5-\left(x-3\right)\left(x-2\right)\times 8=0$$

Use the distributive property to multiply x-3 by x+8 and combine like terms.

$$\left(x^{2}+5x-24\right)\times 3+\left(x-2\right)\left(x+8\right)\times 5-\left(x-3\right)\left(x-2\right)\times 8=0$$

Use the distributive property to multiply x^{2}+5x-24 by 3.

$$3x^{2}+15x-72+\left(x-2\right)\left(x+8\right)\times 5-\left(x-3\right)\left(x-2\right)\times 8=0$$

Use the distributive property to multiply x-2 by x+8 and combine like terms.

$$3x^{2}+15x-72+\left(x^{2}+6x-16\right)\times 5-\left(x-3\right)\left(x-2\right)\times 8=0$$

Use the distributive property to multiply x^{2}+6x-16 by 5.

$$3x^{2}+15x-72+5x^{2}+30x-80-\left(x-3\right)\left(x-2\right)\times 8=0$$

Combine 3x^{2} and 5x^{2} to get 8x^{2}.

$$8x^{2}+15x-72+30x-80-\left(x-3\right)\left(x-2\right)\times 8=0$$

Combine 15x and 30x to get 45x.

$$8x^{2}+45x-72-80-\left(x-3\right)\left(x-2\right)\times 8=0$$

Subtract 80 from -72 to get -152.

$$8x^{2}+45x-152-\left(x-3\right)\left(x-2\right)\times 8=0$$

Use the distributive property to multiply x-3 by x-2 and combine like terms.

$$8x^{2}+45x-152-\left(x^{2}-5x+6\right)\times 8=0$$

Use the distributive property to multiply x^{2}-5x+6 by 8.

$$8x^{2}+45x-152-\left(8x^{2}-40x+48\right)=0$$

To find the opposite of 8x^{2}-40x+48, find the opposite of each term.

$$8x^{2}+45x-152-8x^{2}+40x-48=0$$

Combine 8x^{2} and -8x^{2} to get 0.

$$45x-152+40x-48=0$$

Combine 45x and 40x to get 85x.

$$85x-152-48=0$$

Subtract 48 from -152 to get -200.

$$85x-200=0$$

Add 200 to both sides. Anything plus zero gives itself.

$$85x=200$$

Divide both sides by 85.

$$x=\frac{200}{85}$$

Reduce the fraction \frac{200}{85} to lowest terms by extracting and canceling out 5.

$$x=\frac{40}{17}$$