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

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

Solve for x

$$x=\frac{9}{13}\approx 0.692307692$$

Steps for Solving Linear Equation

Variable x cannot be equal to any of the values -2,\frac{1}{3} since division by zero is not defined. Multiply both sides of the equation by \left(3x-1\right)\left(x+2\right), the least common multiple of x+2,3x-1.

$$\left(3x-1\right)\left(x-3\right)+\left(x+2\right)\left(3x+1\right)=2\left(3x-1\right)\left(x+2\right)$$

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

$$3x^{2}-10x+3+\left(x+2\right)\left(3x+1\right)=2\left(3x-1\right)\left(x+2\right)$$

Use the distributive property to multiply x+2 by 3x+1 and combine like terms.

$$3x^{2}-10x+3+3x^{2}+7x+2=2\left(3x-1\right)\left(x+2\right)$$

Combine 3x^{2} and 3x^{2} to get 6x^{2}.

$$6x^{2}-10x+3+7x+2=2\left(3x-1\right)\left(x+2\right)$$

Combine -10x and 7x to get -3x.

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

Add 3 and 2 to get 5.

$$6x^{2}-3x+5=2\left(3x-1\right)\left(x+2\right)$$

Use the distributive property to multiply 2 by 3x-1.

$$6x^{2}-3x+5=\left(6x-2\right)\left(x+2\right)$$

Use the distributive property to multiply 6x-2 by x+2 and combine like terms.

$$6x^{2}-3x+5=6x^{2}+10x-4$$

Subtract 6x^{2} from both sides.

$$6x^{2}-3x+5-6x^{2}=10x-4$$

Combine 6x^{2} and -6x^{2} to get 0.

$$-3x+5=10x-4$$

Subtract 10x from both sides.

$$-3x+5-10x=-4$$

Combine -3x and -10x to get -13x.

$$-13x+5=-4$$

Subtract 5 from both sides.

$$-13x=-4-5$$

Subtract 5 from -4 to get -9.

$$-13x=-9$$

Divide both sides by -13.

$$x=\frac{-9}{-13}$$

Fraction \frac{-9}{-13} can be simplified to \frac{9}{13} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{9}{13}$$