Solve 4/x-1-2/x-3=2/x+3

$$\frac{ 4 }{ x-1 } - \frac{ 2 }{ x-3 } = \frac{ 2 }{ x+3 }$$

Solve for x

$$x=9$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -3,1,3 since division by zero is not defined. Multiply both sides of the equation by \left(x-3\right)\left(x-1\right)\left(x+3\right), the least common multiple of x-1,x-3,x+3.

$$\left(x^{2}-9\right)\times 4-\left(x-1\right)\left(x+3\right)\times 2=\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply x^{2}-9 by 4.

$$4x^{2}-36-\left(x-1\right)\left(x+3\right)\times 2=\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply x-1 by x+3 and combine like terms.

$$4x^{2}-36-\left(x^{2}+2x-3\right)\times 2=\left(x-3\right)\left(x-1\right)\times 2$$

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

$$4x^{2}-36-\left(2x^{2}+4x-6\right)=\left(x-3\right)\left(x-1\right)\times 2$$

To find the opposite of 2x^{2}+4x-6, find the opposite of each term.

$$4x^{2}-36-2x^{2}-4x+6=\left(x-3\right)\left(x-1\right)\times 2$$

Combine 4x^{2} and -2x^{2} to get 2x^{2}.

$$2x^{2}-36-4x+6=\left(x-3\right)\left(x-1\right)\times 2$$

Add -36 and 6 to get -30.

$$2x^{2}-30-4x=\left(x-3\right)\left(x-1\right)\times 2$$

Use the distributive property to multiply x-3 by x-1 and combine like terms.

$$2x^{2}-30-4x=\left(x^{2}-4x+3\right)\times 2$$

Use the distributive property to multiply x^{2}-4x+3 by 2.

$$2x^{2}-30-4x=2x^{2}-8x+6$$

Subtract 2x^{2} from both sides.

$$2x^{2}-30-4x-2x^{2}=-8x+6$$

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

$$-30-4x=-8x+6$$

Add 8x to both sides.

$$-30-4x+8x=6$$

Combine -4x and 8x to get 4x.

$$-30+4x=6$$

Add 30 to both sides.

$$4x=6+30$$

Add 6 and 30 to get 36.

$$4x=36$$

Divide both sides by 4.

$$x=\frac{36}{4}$$

Divide 36 by 4 to get 9.

$$x=9$$